QUESTON 1).
The angle between a pair of tangents drawn from a point P to the circle x² + y² + 4x - 6y + 9 sin² $\alpha$ + 13cos² $\alpha$ = 0 is 2 $\alpha$ . The equation of the locus of the point P is
1. x² + y² + 4x - 6y + 4 = 0
2. x² + y² + 4x - 6y – 9 = 0
3. x² + y² + 4x - 6y - 4 = 0
4. x² + y² + 4x - 6y + 9 = 0

Question

QUESTON 1).

The angle between a pair of tangents drawn from a point P to the circle x² + y² + 4x - 6y + 9 sin² $\alpha$ + 13cos² $\alpha$ = 0 is 2 $\alpha$ . The equation of the locus of the point P is

1. x² + y² + 4x - 6y + 4 = 0

2. x² + y² + 4x - 6y – 9 = 0

3. x² + y² + 4x - 6y - 4 = 0

4. x² + y² + 4x - 6y + 9 = 0

navneet chandan · Answer

The equation of the circle is x 2 + y 2 + 4x - 6y + 9 sin 2 +13 cos 2 = 0 Hence, the center of the circle = (-2, 3) Radius of the circle is v (-2) 2 + (3 2 ) – 9 sin 2 -13cos 2 = v13 – 9 sin 2 -13cos 2 = v13sin 2 – 9 sin 2 = v4 sin 2 = 2 sin Now, sin = OA/OP = 2 sin / v(h+2) 2 + (k-3) 2 Hence, (h+2) 2 + (k-3) 2 = 4 h 2 + k 2 + 4h -6k +9 = 0. Therefore, the locus of P is x 2 +y 2 + 4x - 6y +9 = 0.

navneet chandan · Answer

image related to the above solution. For futhur queries, mail me on navneet25799@gmail. com Navneet Chandan IIT DELHI

ankit singh · Answer

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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QUESTON 1).The angle between a pair of tangents drawn from a point P to the circle x2 + y2 + 4x - 6y + 9 sin2 + 13cos2 = 0 is 2. The equation of the locus of the point P is1. x2 + y2 + 4x - 6y + 4 = 02. x2 + y2 + 4x - 6y – 9 = 03. x2 + y2 + 4x - 6y - 4 = 04. x2 + y2 + 4x - 6y + 9 = 0