Coordinate Geometry
SYNOPSIS
Coordinate Geometry Coordination of Algebra and Geometry is called coordinate geometry. 1. Cartesian Coordinate System: XOX’ and YOY’ are coordinate axes. The axes divide the coordinate system into four regions called quadrants. x y 1st quadrant + + 2nd quadrant - + 3rd quadrant - - 4th quadrant + -
2. Lattice point: A point whose abscissa and ordinate are integers.
3. Distance between two points P(x1, y1) and Q(x2, y2) is given by
22 PQ = ()()x2 x 1 y 2 y 1
4. Coordinates of different centres of a Triangle:
Centroid: Circumcentre: The point of concurrency (intersection) of A point which is equidistant from all the the three
vertices of a triangle. medians of a triangle.
www.topperlearning.com 1 In-centre: The point of concurrency (intersection) of the internal bisectors of the angles of a triangle.
Ex-centre: Orthocentre: The point at which the bisector of one The point of concurrency (intersection) of interior angle meets the lines bisecting the altitudes of a triangle. the two external angles of the opposite side.
5. Locus and its equation: i. Locus is the curve described by a point which moves under the given condition(s). ii. Equation of the locus of a point is the relation which is satisfied by the coordinates of every point on the locus of the point.
6. Polar coordinates: Polar coordinates express the location of a point as (r, ), where r → the distance of a point from the origin Ө → the angle from the positive x-axis to the point
7. Shifting of origin: Shifting the origin to another point by drawing two lines, one parallel to the x-axis and another parallel to the y-axis is the translation of axes. Intersection of two lines drawn is the origin for the new coordinate system or of translation.
8. Rotation of Axes: In the rotation of axes, the origin is kept fixed whereas the X and Y axes are to be rotated to obtain the new coordinate axes X’ and Y’.
www.topperlearning.com 2 9. Slope of a line: The angle made by the line with the positive direction of the x-axis and measured anticlockwise is called the inclination of the line. The trigonometric tangent of this angle is called the slope (gradient) of the line.
10. Collinearity of Three points: Three points A, B and C are collinear if the slope of line AB is equal to the slope of line BC.
11. Angle between two lines:
If two lines L1 and L2 are parallel, then the angle between them is 0°.
If two lines L1 and L2 are perpendicular, then the angle between them is 90°.
Straight Lines
A line parallel to the y-axis will be of the form x = a, where ‘a’ is the distance between the line and y- axis.
12. A line parallel to the x-axis will be of the form y = b, where ‘b’ is the distance between the line and x- axis.
13. Intercepts of a line: If a line L cuts the x-axis at point A(a, 0) and the y-axis at point B(0, b), then a and b are its x- intercept and y-intercept, respectively.
14. Concurrency of three lines: Three lines are said to be concurrent if they pass through a common point (meet at only one point).
15. Family of Straight Lines: Set of infinite straight lines which pass through (intersect at) a single point A.
16. General form of the second degree equation in x and y: General form of second degree equation in x and y is given by ax2 + 2hxy + by2 +2gx + 2fy + c = 0
17. Homogeneous equation of nth degree: An equation (whose RHS is zero) in which the sum of the powers of x and y in every term is the same (say n) is called a homogeneous equation of nth degree.
Circle
1. Circle is a locus of a point which moves in a plane such that its distance from a fixed point is always constant. A fixed point is the centre of the circle.
2. Parts of a Circle: www.topperlearning.com 3 i. Circumference: Length of the boundary or outer edge of the circle.
ii. Radius: Length of a line from the centre to the edge of the circle.
iii. Diameter: Length of a line which passes through the centre with its endpoints lying on the circle.
iv. Chord: A straight line joining any two points lying on the circumference of a circle.
v. Arc: A part of the circumference of a circle.
vi. Sector: The area which is enclosed by an arc and the two radii of a circle.
vii. Segment: The area inside a circle which is enclosed by an arc and the chord.
viii. Tangent: A straight line which touches the circle at a point.
3. Normal to a Circle The normal of a circle at any point is a straight line which is perpendicular to the tangent at the point of contact. Note: Normal of the circle always passes through the centre of the circle.
4. Chord of Contact From any external point A, draw a pair of tangents touching the circle at points P and Q. Then, PQ is the chord of contact with P and Q as its points of contact.
5. Director Circle of a Circle: The locus of the point of intersection of two perpendicular tangents to a given circle is called its director circle.
www.topperlearning.com 4 6. Angle of Intersection of two Circles: Angle between two circles is defined as the angle between the tangents of the two circles at the point of intersection.
7. Orthogonal Circles: If the angle between the circles is 90°, then the circles are said to be orthogonal circles. We can also say that they cut each other orthogonally. 8. Radical Axis: Radical axis of two circles is the locus of the point which moves such that the lengths of the tangents drawn from it to the two circles are equal.
9. Radical centre: Point at which the radical axes of three circles taken in pairs meet.
10. Common chord: Chord joining the points of intersection of two circles.
Parabola
The section obtained by the intersection of a plane with a cone is called a conic section. 1. Parabola: A symmetrical open plane curve obtained by the intersection of a cone with a plane parallel to its side (base). General Equation of a parabola: 푦2 = 4푎푥
2. Recognising conics: General equation of conics: 푎푥2 + 2ℎ푥푦 + 푏푦2 + 2푔푥 + 2푓푦 + 푐 = 0 훥 = 푎푏푐 + 2푓푔ℎ − 푎푓2 − 푏푔2 − 푐ℎ2
Condition Nature of Conics Δ≠ 0, h = 0, a = b Circle Δ ≠ 0, ab h2 = 0 Parabola Δ ≠ 0, ab h2 > 0 Ellipse or empty set Δ ≠ 0, ab h2 < 0 Hyperbola Δ ≠ 0, ab h2 < 0 and a + b = 0 Rectangular hyperbola
3. Parameters of a parabola: Vertex: (0, 0) Focus: (a, 0)
Axis: y = 0 (X-axis) Directrix: x = a
Focal Distance: Focal Chord: Distance of a point on the parabola from A chord which passes through the focus. the focus. Latus Rectum: Double ordinate passing through the www.topperlearning.com 5 Double Ordinate: focus. A chord which is perpendicular to the Length of the latus rectum is 4a. axis of symmetry.
4. Types of Parabola: Horizontal Parabola
Vertical Parabola
5. Tangent, Normal and Chord to a Parabola Tangent to a Parabola Pair of tangents to a Parabola
www.topperlearning.com 6 Normal to a Parabola Chord of a Parabola Chord of Contact
6. Reflection Property of a Parabola: The tangent at any point P to a parabola bisects the distance between the focal chord through P and the perpendicular from P to the directrix.
Ellipse 1. Ellipse: i. Ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant. ii. The two fixed points are called the foci of the ellipse. iii. The midpoint of the line joining the two foci is called the centre of the ellipse.
2. Parameters of an Ellipse: Vertices are A and A’ Minor axis: BB’ Major axis: AA’ a Focal Radii: SP and S’P b2 Directrices: x Eccentricity: e1, a b e Focal distance: Sum of the a2 Focal chord: A chord focal radii of any point is Double ordinate: A chord which passes through a equal to the length of the perpendicular to the major axis. focus. major axis. 2푏2 Latus Rectum: Length = 푎
www.topperlearning.com 7 3. Auxiliary Circle and Eccentric angle: A circle described on the major axis as a diameter is called the Auxiliary circle. Equation of Auxiliary circle: 푥2 + 푦2 = 푎2 Take two points P and Q on the ellipse and auxiliary circle respectively, such that the x-coordinate is the same for both points. Here, Ө is called the eccentric angle of point P.
4. Tangent, normal and chord of an ellipse:
Tangent to an Ellipse Pair of tangents to an Ellipse
P P
Normal to an Ellipse Chord of Parabola Chord of Contact
P P P Chord of Contact Q Q
5. Director Circle of an Ellipse: 90 Locus of the point of intersection of the tangents which meet at right angles is called the Director circle. The Director circle is given as x2 y 2 a 2 b 2.
www.topperlearning.com 8 Hyperbola
1. Hyperbola: A hyperbola is the set of all points, the difference of whose distance from two fixed points is constant.
2. Parameters of a Hyperbola:
Foci: The two fixed points are called Centre (C): The midpoint of the line joining the foci of the Hyperbola. the foci. Transverse axis: The line through Conjugate axis: The line through the the foci. centre and perpendicular to the Transverse Eccentricity: axis. Conjugate Axis b2 Focal distance: Distance of any point on (e) = 11 Transverse Axis a2 the hyperbola from the foci. Vertices: The points at which the |PS PS’| = 2a and Focal length SS’ = 2ae hyperbola intersects the transverse Distance between foci = 2c a axis. Equations of Directrices: x A(a, 0) and A’(a, 0) e Foci S = (ae, 0) and S’ = (ae, 0)
3. Tangent and Normal to the Hyperbola: Tangent to the Hyperbola Normal to the Hyperbola
www.topperlearning.com 9
4. Director Circle of a Hyperbola: Locus of the point of intersection of tangents which are at right angles is called the director circle. It is given by 푥2 + 푦2 = 푎2 − 푏2
5. Conjugate Hyperbola: For a hyperbola, there exist a hyperbola such that the conjugate and transverse axes of one is equal to the conjugate and transverse axes of the other. Such hyperbolas are known as conjugate to each other. 푥2 푦2 푥2 푦2 E.g.: − = 1 and − = −1 푎2 푏2 푎2 푏2 6. Rectangular (equilateral) hyperbola: The length of the transverse and conjugate axis are equal for a Rectangular hyperbola. i. Vertices: (c, c) & (c, c) ii. Directrices: x + y = 2c iii. Foci: 2c, 2 & 2c, 2c iv. Latus Rectum: ℓ = 2 2 c v. Eccentricity: e = 2
7. Asymptotes of Hyperbola: i. Asymptotes are the lines which get closer and closer to the curve but never touch the curve. ii. When we combine the equation of 푥2 푦2 asymptotes, it gives − = 0 푎2 푏2 iii. So, the equation of a pair of asymptotes differ with a hyperbola and conjugate hyperbola by the same constant only. iv. Asymptotes are the tangents to hyperbola with the centre at infinity. v. They pass through the centre of the hyperbola and the bisectors of angles between them are the axes of the hyperbola.
www.topperlearning.com 10 8. Auxiliary Circle and eccentric angle: A circle drawn with centre C and diameter as transverse axis is called the auxiliary circle of the hyperbola. Its equation is 푥2 + 푦2 = 푎2 is the eccentric angle of the point P on the hyperbola.
www.topperlearning.com 11