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 $AB$ are equal, is
(1) $9x^2+y^2-6xy-54x-62y+241=0$
(2) $x^2+9y^2+6xy-54x+62y-241=0$
(3) $9x^2+9y^2-6xy-54x-62y-241=0$
(4) $x^2+y^2-2xy+27x+31y-120=0$

• A. $9x^2+y^2-6xy-54x-62y+241=0$
• B. $x^2+9y^2+6xy-54x+62y-241=0$
• C. $9x^2+9y^2-6xy-54x-62y-241=0$
• D. $x^2+y^2-2xy+27x+31y-120=0$

Answer 1

Answer: (D)

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