Precalculus
Rotation of Conics
Identify the equation without completing the square. 1) 4y2 - 3x + 2y = 0 1) A) hyperbola B) ellipse C) parabola D) not a conic
Determine the appropriate rotation formulas to use so that the new equation contains no xy -term. 2) x2 + 2xy + y2 - 8x + 8y = 0 2) 2 2 A) x = (xʹ - yʹ) and y = (xʹ + yʹ) 2 2 1 3 3 1 B) x = xʹ - yʹ and y = xʹ + yʹ 2 2 2 2 2 + 2 2 - 2 2 - 2 2 + 2 C) x = xʹ - yʹ and y = xʹ + yʹ 2 2 2 2 D) x = -yʹ and y = xʹ
Rotate the axes so that the new equation contains no xy -term. Discuss the new equation. 3) x2 + 2xy + y2 - 8x + 8y = 0 3) A) θ = 36.9° B) θ = 45° xʹ2 yʹ2 yʹ2 = -42xʹ + = 1 4 4 parabola ellipse vertex at (0, 0) center (0, 0) focus at (- 2, 0) major axis is xʹ-axis vertices at (±2, 0) C) θ = 45° D) θ = 36.9° xʹ2 = -42yʹ xʹ2 yʹ2 + = 1 parabola 4 2 vertex at (0, 0) ellipse focus at (0, - 2) center (0, 0) major axis is xʹ-axis vertices at (±2, 0)
Identify the equation without applying a rotation of axes. 4) x2 + 12xy + 36y2 - 4x + 3y - 10 = 0 4) A) ellipse B) parabola C) hyperbola D) not a conic
5) 2x2 + 6xy + 9y2 - 3x + 2y + 6 = 0 5) A) hyperbola B) ellipse C) parabola D) not a conic
6) 3x2 + 12xy + 2y2 - 3x - 2y + 5 = 0 6) A) ellipse B) hyperbola C) parabola D) not a conic
Precalculus 7) x2 + 3xy - 2y2 + 4x - 4y + 1 = 0 7) A) hyperbola B) parabola C) ellipse D) not a conic
8) 5x2 - 3xy + 2y2 + 3x + 4y + 2 = 0 8) A) hyperbola B) ellipse C) parabola D) not a conic
Rotate the axes so that the new equation contains no xy -term. Discuss the new equation. 9) 31x2 + 10 3xy + 21y2 -144 = 0 9) A) θ = 36.9° B) θ = 30° xʹ2 yʹ2 xʹ2 yʹ2 + = 1 + = 1 9 4 4 9 ellipse ellipse center at (0, 0) center at (0, 0) major axis is xʹ-axis major axis is yʹ-axis vertices at (±3, 0) vertices at (0, ±3) C) θ = 45° D) θ = 45° xʹ2 = -42yʹ yʹ2 = -42xʹ parabola parabola vertex at (0, 0) vertex at (0, 0) focus at (0, - 2) focus at (- 2, 0)
10) xy +16 = 0 10) A) θ = 45° B) θ = 45° yʹ2 xʹ2 yʹ2 = -32xʹ - = 1 32 32 parabola hyperbola vertex at (0, 0) center at (0, 0) focus at (-8, 0) transverse axis is yʹ-axis vertices at (0, ±42) C) θ = 36.9° D) θ = 45° xʹ2 yʹ2 yʹ2 xʹ2 + = 1 + = 1 4 2 32 32 ellipse ellipse center at (0, 0) center at (0, 0) major axis is the xʹ-axis major axis is yʹ-axis vertices at (±2, 0) vertices at (0, ±42)
Precalculus 11) x2 + xy + y2 - 3y - 6 = 0 11) A) θ = 45° B) θ = 45° xʹ2 yʹ2 yʹ2 = -18xʹ - = 1 6 8 parabola hyperbola vertex at (0, 0) 9 center at (0, 0) focus at (- , 0) transverse axis is the xʹ-axis 2 vertices at (± 6, 0)
C) θ = 45° D) θ = 45° xʹ2 yʹ2 2 2 322 + = 1 xʹ - yʹ - 3 4 2 2 + = 1 ellipse 5 15 center at (0, 0) ellipse major axis is yʹ-axis 2 32 center at ( , ) vertices at (0, ±2) 2 2 major axis is yʹ-axis 2 32 2 92 vertices at ( , - ) and ( , ) 2 2 2 2
12) 5x2 - 6xy + 5y2 - 8 = 0 12) A) θ = 45° B) θ = 45° yʹ2 = -4xʹ xʹ2 + yʹ2 = 1 parabola 4 vertex at (0, 0) ellipse focus at (-1, 0) center at (0, 0) major axis is the xʹ-axis vertices at (±2, 0) C) θ = 45° D) θ = 45° xʹ2 = -4yʹ xʹ2 - yʹ2 = 1 parabola 4 vertex at (0, 0) hyperbola focus at (0, -1) center at (0, 0) transverse axis is the xʹ-axis vertices at (±2, 0)
Determine the appropriate rotation formulas to use so that the new equation contains no xy -term. 13) 4x2 + 2xy + 4y2 - 8x + 8y = 0 13) A) x = -yʹ and y = xʹ 1 3 3 1 B) x = xʹ - yʹ and y = xʹ + yʹ 2 2 2 2 2 2 C) x = (xʹ - yʹ) and y = (xʹ + yʹ) 2 2 2 + 2 2 - 2 2 - 2 2 + 2 D) x = xʹ - yʹ and y = xʹ + yʹ 2 2 2 2
Precalculus 14) 9x2 - 4xy + 5y2 - 8x + 8y = 0 14) A) x = -yʹ and y = xʹ 2 2 B) x = (xʹ - yʹ) and y = (xʹ + yʹ) 2 2 1 3 3 1 C) x = xʹ - yʹ and y = xʹ + yʹ 2 2 2 2 2 - 2 2 + 2 2 + 2 2 - 2 D) x = xʹ - yʹ and y = xʹ + yʹ 2 2 2 2
Identify the equation without completing the square. 15) 4x2 + 3y2 + 7x - 3y = 0 15) A) parabola B) hyperbola C) ellipse D) not a conic
16) 3x2 - 4y2 - 4x + 2y + 1 = 0 16) A) hyperbola B) ellipse C) parabola D) not a conic
Precalculus Answer Key Testname: ROTATION OF CONICS PRACTICE
1) C 2) A 3) C 4) B 5) B 6) B 7) A 8) B 9) B 10) A 11) D 12) B 13) C 14) D 15) C 16) A
Precalculus