The angle between a pair of tangents drawn from a point P to the circle x² + y² + 4x – 6y + 9 sin²α + 13 cos²α = 0 is 2α. The equation of the locus of the point P is
(a) x² + y² + 4x – 6y + 4 = 0
(b) x² + y² + 4x – 6y – 9 = 0
(c) x² + y² + 4x – 6y – 4 = 0
(d) x² + y² + 4x – 6y + 9 = 0
Answers
Ans is D. I hope this will help you.
x² + y² + 4x - 6y + 9 = 0 is the Locus of point P if The angle between a pair of tangents drawn from a point P to the circle x² + y² + 4x – 6y + 9 sin²α + 13 cos²α = 0 is 2α.
Step-by-step explanation:
x² + y² + 4x – 6y + 9 sin²α + 13 cos²α = 0
=> x² + 4x + 4 - 4 + y² - 6y + 9 - 9 + 9 sin²α + 13 cos²α = 0
=> (x +2)² +(y - 3)² - 13 + 9 sin²α + 13 cos²α = 0
=> (x +2)² +(y - 3)² = 13 - 13 cos²α - 9 sin²α
=> (x +2)² +(y - 3)² = 13Sin²α - 9 sin²α
=> (x +2)² +(y - 3)² = 4Sin²α
=> (x +2)² +(y - 3)² = (2Sinα)²
center = -2 , 3
radius = 2 Sinα
The angle between a pair of tangents drawn from a point P = 2α
half of 2α = α
=> Sinα = Radius/(distance of P from center of Circle)
=> distance of P from center of Circle = Radius/Sinα
=> distance of P from center of Circle = 2 Sinα/Sinα
=> distance of P from center of Circle = 2
Distance of P from center
= √(x - (-2))² + (y - 3)² = 2
Squaring both sides
=> x² + 4x + 4 + y² - 6y + 9 = 4
=> x² + y² + 4x - 6y + 9 = 0
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