Question

Tangents are drawn from the point $P(3,4)$ to the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ touching the ellipse at points $A$ and $B$.
The equation of the locus of the point whose distance from the point $P$ and the line $A B$ are equal, is
(1) $9 x^{2}+y^{2}-6 x y-54 x-62 y+241=0$
(2) $x^{2}+9 y^{2}+6 x y-54 x+62 y-241=0$
(3) $9 x^{2}+9 y^{2}-6 x y-54 x-62 y-241=0$
(4) $x^{2}+y^{2}-2 x y+27 x+31 y-120=0$

• A. $ 9x^ 2+ y^2-6 x y - 5 4 x - 6 2 y +241 =0 $
• B. $ x^2 + 9y^2 + 6 x y -5 4 x + 6 2 y -2 4 1 =0 $
• C. $ 9x^2 + 9y^2-6 x y -5 4 x - 6 2 y - 2 4 1 =0 $
• D. $ x ^2+ y^ 2-2 x y + 2 7 x + 3 1 y -1 2 0 = 0 $

Answer 1

Answer: (D)

Equation of $AB$ is $ y-0= -\frac{1}{3} ( x-3)$
$x + 3 y- 3 = 0 $
$ \Rightarrow | x+ 3y -3 | ^2 = 10 [ ( x-3)^2 + (y- 4)^2 ] $
[Look at coefficients of $x^2$ and $y ^2$ in the answers]