1. CONIC SECTIONS : CASE (I) : WHEN THE FOCUS LIES ON THE DIRECTRIX. (i) Vertex is (0, 0) A , or conic is the locus of a point In this case D = abc + 2fgh - af2 - bg2 - ch2 = (ii) focus is (a, 0) which moves in a plane so that its distance 0 & the general equation of a conic (iii) Axis is y = 0 from a fixed point is in a constant ratio to its represents a pair of straight lines if : (iv) Directrix is x + a = 0 distance from a fixed straight line e > 1 the line will be real & distinct ➢ The fixed point is called the FOCUS. intersecting at S. ➢ The fixed straight line is called the FOCAL DISTANCE : The distance of a point on E = 1 the line will coincident. DIRECTRIX. the from the focus is called the e < 1 the lines will be imaginary. ➢ The constant ratio is called the FOCAL DISTANCE OF THE POINT. ECCENTRICITY denoted by e. ➢ The line passing through the focus & FOCAL CHOD : perpendicular to the distance called A chord of the parabola, which passes through CASE (II) : WHEN THE FOCUS DOES NOT the axis the focus is called a FOCAL CHORD. LIE ON DIRECTRIX. ➢ A point of intersection of a conic with A parabola an a its axis is called a VERTEX. DOUBLE ORDINATE : A chord of the parabola rectangular hyperbola perpendicular to the axis of the symmetry is C = 1 ; D ≠ 0, 0 < e < 1 ; D ≠ 0; e > 1 ; D ≠ 0; e called a DOUBLE ORDINATE. > 1 ; D ≠ 0 h2 = ab, h2 < ab, h2 > ab, h2 > ab ; a 2. GENERAL EQUATION OF A CONIC : + b = 0 LATUS RECTUM : FOCAL DIRECTRIX PROPERTY : A double ordinate passing through the focus The general equation of a conic with focus 4. PARABOLA : or a focal chord perpendicular to the axis of (p, q) % directrix lx + my + n = 0 is : A parabola is the locus of a point which parabola is called the LATUS RECTUM. For (l2 + m2)[(x - p)2 + (y - q)2] = e2(lx + my + n)2 = moves in a plane, such that its distance from Y2 = 4ax. ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 a fixed point (focus) is equal to its Length of the latus rectum = 4a. perpendicular distance from a fixed straight Ends of the latus rectum are L(a,2a) & L’(a,-2a). 3. DISTINGUISHING BETWEEN THE CONIC : line (directrix). Note that : The nature of the conic section depends upon Standard equation of the parabola is y2 = (i) perpendicular distance from focus on the position of the focus S w.r.t. The directrix & 4ax. For this parabola : directrix = half the latus rectum. also upto the value of the eccentricity e. Two

different cases arise. (ii) Vertex is middle point of the focus & the (c) If the normals of the parabola y2 = 4ax at the point of intersection of directrix & axis. Note : If chord joining t t & t t pass a 1 2 3 4 points t & t intersect again on the parabola at (iii) Two are laid to be equal if they through point (c,0) on axis, then t t = t t = -c/a. 1 2 1 2 3 4, the point ‘t ’ then t t = 2; t = -(t + t ) and the have the same latus rectum. Four standard 3 1 2 3 1 3 line joining t & t passes through a fixed point forms of the parabola are y2 = 4ax ; y2 = - 4ax ; 9. TO THE PARABOLA y2 = 4ax : 1 2 2 2 (-2a, 0). X = 4ay; x = - 4ay (i) yy1 = 2a (x + x1 ) at the point ( x1 , y1 ) ; (ii) y = mx + (m ≠ o) at General Note : 5. POSITION OF A POINT RELATIVE TO A (iii) ty = x + at2 at (at2, 2at). (i) Length of subtangent at any point P(x, y) on PARABOLA : Note : Point of intersection of the tangents at the parabola y2 = 4ax equals twice the abscissa The point ( x , y ) lies outside, on or inside the 1 1 the point t & t is [ at t a(t + t ) ]. of the point P. Note that the subtangent is parabola y2 = 4ax according as the expression 1 2 1 2 1 2 bisected at the vertex. y 2 - 4ax is positive zero or negative. 1 1 10. NORMALS TO THE PARABOLA y2 = 4ax (ii) Length of subnormal is constant for all : points on the parabola & is equal to the semi 6. LINE & A PARABOLA : (i) y - y = latus rectum. The line y = mx + c meets the parabola y2 = 4ax 1 (ii) y = mx - 2m - am3 at (am2 - 2am) (iii) If a family of straight lines can be in two points real, coincident or imaginary (iii) y + tx = 2at + at3 at (at2, 2at). represented by an equation λ2P + λQ + R = 0 according as a ≷ cm ⇒ condition of tangency where λ is a parameter and P, Q, R are linear is, Note : Point of intersection of normals at, t & t 1 3 functions of x and y then the family of lines will are, a(t 2 + t 2 + t 1 2 1 be to the curve Q2 = 4PR. 7. Length of the chord intercepted by the parabola on the line y = mx + c is : 11. THREE VERY IMPORTANT RESULTS :

(a) If t1 & t2 are the ends of a focal chord 12. The equation to the pair of tangents which of the parabola y2 = 4ax then t t = -1. 8. PARAMETRIC REPRESENTATION : 1 2 can be drawn from any point (x1, y1) to the Hence the coordinates at the 2 2 The simplest & best form of representing the parabola y = 4ax is given by : SS1 = T where : 2 extremities of a focal chord can be 2 2 coordinates of a point on the parabola is (at , S ≡ y - 4ax ; S1 ≡ y1 - 4ax1 ; T ≡ yy1 - 2a( x + x1). 2at). The equations X = at2 & y = 2at together taken as (at2, 2at ) & represents the parabola y2 = 4ax, t being the (b) If the normals to the parabola y2 = 4ax at the point t , meets the parameter. The equation of a chord joining t1 & 1 parabola again at the point t , then t2 is 2x - (t1 + t2)y + 2 at t1t2 = 0. 2 13.DIRECTOR : The tangent at the point. Locus of the point of intersection of the (iii) A line segment from a point P on the (iii) If the polar of a point P passes through the perpendicular tangents to the parabola y2 = parabola and parallel to the system of parallel point Q, then the polar of Q goes through P. 4ax is called the DIRECTOR CIRCLE. It’s chords is called the ordinate to the diameter (iv) Two straight lines are said to be conjugated equation is x + a = 0 which is parabola’s own bisecting the system of parallel chords and the to each other w.r.t. A parabola when the pole of directrix. chords are called its double ordinate. one lies on the other. 14. CHORD OF CONTACT : (v) Plar of a given point P w.r.t. Any Conic is the 18. IMPORTANT HIGHLIGHTS : Equation to the chord of contact of tangent locus of the harmonic conjugate of P w.r.t. The (a) If the tangents & normal at any point ‘P’ drawn from a point P(x , y ) is yy = 2a(x + x ). two points is which any line though P cuts the 1 1 1 1 of the parabola intersect the axis at T & Remember that the area of the triangle formed conic. G then ST = SG = SP where ‘S’ is the by the tangents from the point (x , y ) & the 1 1 16. CHORD WITH A GIVEN MIDDLE POINT : focus. In other words the tangent and chord of contact is (y 2 - 4ax )3/2 ÷ 2a . Also note 1 1 Equation of the chord of the parabola y2 = 4ax the normal at a point P on the parabola that the chord of contact exists only if the point whose middle point is (x , y ) is are the bisector of the angle between P is not inside. 1 1 This reduced to T = S1 the focal radius SP & the perpendicular 2 Where T ≡ yy1 - 2a( x + x1) & S1 ≡ y1 - 4ax1. from P on the directrix. From this we conclude that all rays emanating from 17. DIAMETER : S will become parallel to the axis of the 15. POLAR & POLE : The locus of the middle points of a system of parabola after reflection. (i) Equation of the Polar of the point P(x , y ) 1 1 parallel chords of a Parabola is called a (b) The point of a tangent to a parabola w.r.t. The parabola y2 = 4ax is yy = 2a(x + x ) 1 1 DIAMETER. Equation to the diameter of a cut off between the directrix & the (ii) The pole of the line lx + my + n = 0 w.r.t. The parabola is y = 2a/m, where m = slope of curve subtends a right angle at focus, parabola y2 = 4ax is parabola chords. (c) The tangents at the extremities of a focal chord intersect at right angles on Note : NOTE : the directrix, and hence a circle on any (i) The polar of the form of the parabola is the (i) The tangent at the extremity of a diameter focal chord as diameter touches the directrix. of a parabola is parallel to the system of chords directrix. Also a circle on any focal radii (ii) When the point (x , y ) lies without the 1 1 it bisects. of a point P(at2, 2at) as diameter parabola the equation to its polar is the same (ii) The tangent the ends of any chords of a as the equation to the chord of contact of parabola meet on the diameter which bisects tangents drawn from (x , y ) when (x , y ) is on 1 1 1 1 the chord. the parabola the polar is the same as the Touches the tangent at the vertex and (k) If normal drawn to a parabola passes ELLIPSE intercepts a chord of length on a through a point P(h, k) then normal at the point P. K = mh - 2am - am3 i.e., am3 + m(2a - h) + k = 0. Standard equation of an ellipse referred to its

(d) Any tangent to a parabola & perpendicular Then gives m1 , m2 & m3 = 0 ; m1m2 + m2m3 + principal axes along the coordinate axes is 2 2 2 ⇒ 2 2 on it from the focus meets on the tangent at m3m1 = Where a > b & b = a (1 - e ) a - b 2 2 the vertex. Where m1 m2 & m3 are the slopes of the three = a c . Where e = eccentricity (0 < e < 1). FOCI : (e) If the tangents at P and Q meet in T, then ; concurrent normals. Note that the algebraic S ≡ (a e,0) & S’ ≡ (-a e, 0). ❏ TP and TQ subtended equal angles at sum of the : EQUATIONS OF DIRECTRICES : the focus S. ❏ Slopes of the three concurrent normals

❏ ST2 = SP, SQ & is zero ❏ The triangles SPT and STQ are similar. ❏ Ordinates of the three conormal points VERTICES : (f) Tangents and normals at the extremities of on the parabola is zero. A’ ≡ (-a, 0) & A ≡ (a, 0). the later rectum of a parabola y2 = 4ax ❏ Centroid of the Δ formed by three constitutes a square, their points of intersection co-normal points lies on the x-axis. MAJOR AXIS : being (-a’,0) & (3a’ 0). (l) A circle circumscribing the triangle formed by ’ ’ The line segment A A in which the foci S & S (g) Semi latus rectum of the parabola y2 = 4ax, the three co-normal points passes through the lies is of length 2a & is called the major axis (a> is the harmonic mean between segments of vertex of the parabola and its equation is, 2(x2 b) of the ellipse. Point of intersection of major any focal chord of the parabola is ; + y2) -2(h + 2a)x - ky = 0 axis with directrix is called the foot of the directrix (z). (h) The circle circumscribing the triangle formed by any three tangents to a parabola MINOR AXIS : passes through the focus. the ellipse in the points B’ = (0 , -b ) & B =( 0,b). (i) The orthocentre of any triangle formed by The line segment B’B of length 2b (b < a) is three tangents to a parabola y2 = 4ax lies on called the Minor Axis of the ellipse. the directrix & has the co-ordinates -a,a(t1 + t2 + t3 + t1t2t3). (j) The area of the triangle formed by three points on a parabola is twice the area of the triangle formed by the tangents at these points. PRINCIPLE AXIS : Note : 4. PARAMETRIC REPRESENTATION : The major & minor axis together are called (i) The sum of the focal distance of any point The equation x = a cos θ & y = b sin θ principal axis of the ellipse. on the ellipse is equal to the major Axis. Hence together represent the ellipse CENTRE : distance of focus from the extremity of a minor Where θ is a parameter. Note that if P(θ) = (a The point which bisects every chord of the axis is equal to semi major axis. i.e, BS = CA. cos θ, b sinθ) is on the ellipse then ; Q(θ) = ( a conic drawn through it is called the center of (ii) If the equation of the ellipse is given as cosθ, a sinθ) is on the auxiliary circle. the conic. C ∞ (0, 0) the origin is the ellipse & nothing is mentioned then the rule is to assume that a > b. 5. LINE AND AN ELLIPSE : The line y = mx + c meets the ellipse 2. POSITIVE OF A POINT w.r.t. AN ELLIPSE : in two points real, coincident or imaginary DIAMETER : The point P(x , y ) lies outside, inside or on the according as c2 is < = or > a2m2 + b2 . Hence y = A chord of the conic which passes through the 1 1 ellipse according as : > < or = 0. mx + c is tangent to the ellipse If c2 = center is called a diameter of the conic. a2m2 + b2. The equation to the chord of the FOCAL CHORD : 3. AUXILIARY CIRCLE/ ECCENTRIC ANGLE : ellipse joining two points with eccentric angles A chord which passes through a focus is called A circle decribed on major axis as diameter is α & β is given by a focal chord. called the auxiliary circle. Let Q be a point on the auxiliary circle. 6. TANGENTS : DOUBLE ORDINATE : X2 + y2 = a2 such that PQ produced is (i) is tangent to the ellipse at

A chord perpendicular to the major axis is perpendicular to the x-axis then P & Q are (x1, y1). called a double ordinate. called as the CORRESPONDING on the ellipse (0 ≤ θ ≤ 2π). Note : The figure formed by the tangents at Note that LATUS RECTUM : The focal chord the extremities of latus rectum is rhombus of Hence “ If from each point of a circle perpendicular to the major axis is called the area. perpendicular are drawn up to a fixed diameter latus rectum. Length of the latus rectum (ii) is tangent to the ellipse then the locus of the points dividing these (LL’) = ( distance for all values of m. in a given ratio is an ellipse of from focus to the corresponding directrix) Note that there are two tangents to the ellipse which the given circle is the auxiliary circle”. having the same m, i.e., there are two tangents parallel to any given direction. (iii) is tangents to the ellipse 10. DIAMETER : (v) locus of the mid point of Gg is another at the point (a cosθ, b sinθ). The locus of the middle points of a system of ellipse having the same eccentricity as that of (iv) The eccentric angles of point of contact of parallel chords with slope ‘m’ of an ellipse is a the original ellipse. [where S and S’ are the focii two parallel tangents differ by π. Conversely if straight line passing through the centre of the of the ellipse and T is the point where tangent the difference between the eccentric angles of ellipse, called its diameter and has the at P meet the major axis] two points is p then the tangents at these equation H - 4 The tangent & normal at a point P on the points are parallel. ellipse bisect the external & internal angles (v) Point of intersection of the tangents at the 11. IMPORTANT HIGHLIGHTS : between the focal distance of P. This refers to point α & β is Referring to an ellipse the well known reflection property of the H - 1 If P be any point on the ellipse with S & S’ ellipse which states that rays from one focus 7. NORMALS : as its foci then l(SP) + l(S’P) = 2a. are reflected through other focus & vice-versa.

H - 2 The product of the length’s of the Hence we can deduce that the straight lines (i) Equation of the normal at (x , y ) is 1 1 perpendicular segments from the foci on any joining each focus to the foot of the

tangent to the ellipse is b2 and the feet of perpendicular from the other focus upon the (ii) Equation of the normal at the point (a cosθ, these perpendicular Y, Y’ lie on its auxiliary tangent at any point P meet on the normal PG b sinθ) is ; ax’secθ - by’ cosecθ = (a2 - b2). circle. The tangents at these feet to the and bisects it where G is the point where (iii) Equation of a normal in terms of its slope auxiliary circle meet on the ordinate of P and normal at P meets the major axis. ‘m’ is y = mx - that the locus of their point of intersection is a H - 5 The portion of the tangent to an ellipse 8. DIRECTOR CIRCLE : similar ellipse as that of the original one. Also between the point of contact & the directrix Locus of the point of intersection of the the lines joining centre to the feet of the subtends a right angle at the corresponding tangents which meet at right angles is called perpendicular Y and focus to the point of focus. www.MathsBySuhag.com , the Director Circle. The equation to this locus contact of tangent are parallel. www.TekoClasses.com is x2 + y2 = a2 + b2 i.e, a circle whose centre is H - 3 If the normal at any point P on the ellipse H - 6 The circle on any focal distance as the center of the ellipse & whose radius is the with centre C meet the major & minor axes in G diameter touches the auxiliary circle. length of the line joining the ends of the major & g respectively & if CF be perpendicular upon H - 7 Perpendicular from the centre upon all & minor axis. this normal then chords which join the ends of any 2 9. Chord of contact, pair of tangents, chord (i) PF. PG = b perpendicular diameter of the ellipse are of 2 with a given middle point, pole & polar are to (ii) PF.Pg = a constant length. ’ be interpreted as they are in parabola. (iii) PG.Pg = SP.S P H - 8 If the tangent at the point P of a standard (iv)CG.CT = CS2 ellipse meets the axis in T and t and CY is the perpendicular on it from the centre then, 2 2 (i) T.t PY = a - b HYPERBOLA 2. FOCAL PROPERTY : And (ii) least value of Tt is a + b. The difference of the focal distances of any point on the hyperbola is constant and equal to The HYPERBOLA is a conic whose transverse axis i.e. ||PS| - |PS’|| = 2a. The eccentricity is greater than unity (e > 1). distance SS’ = focal length. 3. CONJUGATE HYPERBOLA : 1. STANDARD EQUATION & DEFINITIONS(S) Two hyperbola such that transverse & Standard equation of the hyperbola is conjugate axes of one hyperbola are Where b2 = a2(e2 - 1) respectively the conjugate & the transverse Or a2e2 = a2 + b2 i.e e2 = 1 + axes of the other are called CONJUGATE

HYPERBOLAS of each other eg. FOCI : Are conjugate S ≡ (ae, 0) & S’ ≡ (-ae, 0). of each. EQUATIONS OF DIRECTRICES : Note : (a) If e & e are the eccentricities of the 1 2 hyperbola & its conjugate then e -2 + e -2 = 1. VERTICES : 1 2 (b) The foci of a hyperbola and its conjugate A ≡ (a, 0) & A’ ≡ (-a, 0). are concyclic and form the vertices of a square l(Latus rectum) = . (c)Two hyperbolas are said to be similar if they Note : l(L.R) = 2e (distance from focus to the have the same eccentricity. corresponding directrix) TRANSVERSE AXIS : The line segment A’ A of 4. RECTANGULAR OR EQUILATERAL length 2a in which the foci S’ & S both be is HYPERBOLA : called the T.A. OF THE HYPERBOLA. The particular kind of hyperbola in which the CONJUGATE AXIS : The line segment B’B lengths of the transverse & conjugate axis are between the two points B’ = ( 0, -b) & B = ( 0, b ) equal is called an EQUILATERAL is called as the C.A. OF THE HYPERBOLA. HYPERBOLA. Note that the eccentricity of the The T.A. & the C.A. of the hyperbola are rectangular hyperbola is and the length of together called the principal axes of the its latus rectum is equal to its transverse or hyperbola. conjugate axis. 5. AUXILIARY CIRCLE : 6. POSITION OF A POINT ‘P’ w.r.t. A (c) can be taken as the A circle drawn with center C & T.A. as a HYPERBOLA : tangent to the hyperbola diameter is called the AUXILIARY CIRCLE of The quantity is positive, zero or Note that there are two parallel tangents the hyperbola.Equation of the auxiliary circle is negative according as the point (x1 , y1 ) lies having the same slope m. x2+y2=a2. within, upon or without the curve. (d) Equation of a chord joining α & β is Note from the figure that P & Q are called the “CORRESPONDING POINTS” On the 7. LINE AND A HYPERBOLA : The straight line hyperbola & the auxiliary circle. ‘Θ’ is called the y = max + c is a secant, a tangent or passes NORMALS : eccentric angle of the point ‘P’ on the outside the hyperbola according as : (a) The equation of the normal to the hyperbola. (0 ≤ θ ≤ 2π). c2 > = < a2m2 - b2. hyperbola at the point P(x , y ) 1 1 on it is = a2e2. 8. TANGENTS AND NORMALS : TANGENTS : (b) The equation of the normal at the Note : The equations x = a secθ & y = b tanθ (a) Equation of the tangent to the point P ( a secθ, b tanθ) on the together represents the hyperbola hyperbola at the point (x , y ) 1 1 hyperbola is Where θ is a parameter. The parametric is (c) Equation to the chord of contact, polar, equations : x = a cos hΦ, y = b sin hΦ also NOTE : In general two tangent can be drawn chord with a given middle point, pair of represents the same hyperbola. from an external point (x , y ) to the hyperbola 1 1 tangents from an external point is to and they are y - y = m ( x - x ) & y - y = m (x - 1 1 1 1 2 be interpreted as in ellipse. General Note : Since the fundamental x2), where m1 & m2 are roots of the equation 2 2 2 2 2 equation to the hyperbola only differs from that (x1 - a )m - 2x1y1m + y1 + b = 0. If D < 0, then 2 2 to the ellipse in having -b instead of b it will no tangent can be drawn from (x1, y1) to the be found that many propositions for the hyperbola. hyperbola are derived from those for the (b) Equation of the tangent to the ellipse by simply changing the sign of b2. hyperbola at the point ( a secፀ, b tanθ) is

Note : Point of intersection of the tangents at

θ1 & θ2 is 9. DIRECTOR CIRCLE : Hyperbola have the same foci. They cut at 11. ASYMPTOTES : Definition : If the length of The locus of the intersection of tangents which right angles at any of their common point. the perpendicular let fall from a point on a are at right angles is known as the DIRECTOR Note that the ellipse and the hyperbola to a straight line tends to zero as the CIRCLE of the hyperbola. The equation to the hyperbola X are confocal point on the hyperbola moves to infinity along director circle is : x2 + y2 = a2 - b2. and therefore orthogonal. the hyperbola, then the straight line is called If b2 < a2 this circle is real ; if b2 = a2 the radius H - 4 The foci of the hyperbola and the point P the Asymptote of the hyperbola. of the circle is zero & it reduces to a point and Q in which any tangent meets the To find the asymptote of the hyperbola : circle at the origin. In this case the centre is the tangents at the vertices are concyclic with PQ Let y = mx + c is the asymptote of the only point from which the tangents at right as diameter of the circle. hyperbola angles can be drawn to the curve. If b2 > a2, Solving these two we get the quadratic as the radius of the circle is imaginary, so that (b2 - a2m2) x2 - 2a2mcx - a2(b2 + c2) = 0 ……….(1) there is no such circle & so no tangents at right In order that y = mx + c be an asymptote, both angle can be drawn to the curve. roots of equation (1) must approach infinity, the conditions for which are : 10. HIGHLIGHTS ON TANGENT AND Coeff of x2 = 0 & coeff of x = 0. NORMAL : ⇒ b2 - a2m2 = 0 or m = H - 1 Locus of the feet of the perpendicular A2mc = 0 ⇒ c = 0. drawn from focus of the hyperbola ∴ equations of asymptote are and upon any tangent is its auxiliary circle i.ie., x2 + y2 = a2 & the product of the feet of these Combined equation to the asymptotes perpendiculars is b2. (semi C. A)2 H - 2 The portion of the tangent between the point of contact & the directrix subtends a right PARTICULAR CASE : angle at the corresponding focus. When b = a the asymptotes of the rectangular H - 3 The tangents & normal at any point of a hyperbola. X2 - y2 = a2 are, y = ± x which are at hyperbola bisect the angle between the focal right angles. radii. This spells the reflection property of the hyperbola as “ An incoming light ray” aimed towards one focus is reflected from the outer surface of the hyperbola towards the other focus. It follows that if an ellipse and a 12. HIGHLIGHTS ON ASYMPTOTES : Note : (i) Equilateral hyperbola ⇔ rectangular H - 1 If from any point on the asymptote a 13. RECTANGULAR HYPERBOLA : hyperbola straight line be drawn perpendicular to the Rectangular hyperbola referred to its (ii) If a hyperbola is equilateral then the transverse axis, the product of the segments of asymptotes as axis of coordinates conjugate hyperbola is also equilateral. this line, intercepted between the point & the (a) Eq is xy = c2 with parametric representation (iii) A hyperbola and conjugate have the same curve is always equal to the square of the semi x = ct, y = c/t, asymptote. conjugate axis. (iv) The equation of the pair of asymptotes H - 2 Perpendicular from the foci on either (b) Eq. of a chord joining the points (t ) & (t ) is x differ the hyperbola & the conjugate hyperbola 1 2 asymptote meet it in the same points as the + t t y = c(t + t ) with slope by the same constant only. 1 2 1 2 corresponding directrix & the common points (v) The asymptotes pass through the centre of of intersection lie on the auxiliary circle. (c) Equation of the tangents at P(x , y ) is the hyperbola & the bisectors of the angles 1 1 H - 3 The tangents at any point P on a between the asymptotes are the axes of the hyperbola with centre C, meets the hyperbola. www.MathsBySuhag.com , asymptotes in Q and R and cuts off a ΔCQR of (d) Equation of normal : www.TekoClasses.com constant area equal to ab from the asymptotes (vi) The asymptotes of a hyperbola are the & the portion of the tangent intercepted (e) Chord with a given middle point as (h, k) is diagonals of the rectangle formed by the lines between the asymptotes is bisected at the kx + hy = 2hk. drawn through the extremities of each axis point of constant. This implies that locus of the parallel to the other axis. centre of the circle circumscribing the ΔCQR in (vii) Asymptotes are the tangent to the case of a rectangular hyperbola is the hyperbola from the centre. hyperbola itself & for a standard hyperbola the (viii) A simple method to find the coordinates locus would be the curve, of the centre of the hyperbola expressed as a 4(a2x2 - b2y2) = (a2 + b2)2. general equation of degree 2 should be H - 4 If the angle between the asymptotes of a remembered as : hyperbola is 2θ then e = secθ. Let f (x, y) = 0 represents a hyperbola. Find Then the point of intersection of Gives the centre of the hyperbola. Batch Started on: Features th 15 JUNE 2020

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