Paragraph: Tangents are drawn from the point $P(3,4)$ to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$…
Paragraph:
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 $B$.
Question:
The equation of the locus of the point whose distance from the point $P$ and the line $A B$ are equal, is
$9 x^2+y^2-6 x y-54 x-62 y + 241=0$
$x^2+9 y^2+6 x y-54 x+62 y - 241=0$
$9 x^2+9 y^2-6 x y-54 x-62 y - 241=0$
$x^2+y^2-2 x y+27 x+31 y - 120=0$
Solution
Equation of $A B$ is $y-0=-\frac{1}{3}(x-3)$
$
\begin{gathered}
x+3 y-3=0 \\
|x+3 y-3|^2=10\left[(x-3)^2+(y-4)^2\right]
\end{gathered}
$
(Look at coefficient of $x^2$ and $y^2$ in the
answers)