Question

If line $x+y=3$ is a tangent to the ellipse with foci at $(4,3)$ and $(6,k)$ at point $(1,2),$ then the value of $k$ is

• A. $27$
• B. $17$
• C. $20$
• D. $15$

Answer 1

The correct option is B $17$

Let $P(1,2)$ and from reflection property given tangent line is angular bisector of $\angle APB$
So, image of $B$ w.r.t tangent line lies on line AP

Image of $(4,3)$ in the line $x+y-3=0$ is :
$\frac{x-4}{1}=\frac{y-3}{1}=\frac{-2(4+3-3)}{2}=-4$
$\Rightarrow x=0$ and $y=-1$
Now points $(0,-1),(1,2)$ and $(6,k)$ are collinear.
$\Rightarrow \frac{-1-2}{0-1}=\frac{k-2}{6-1}$
$\Rightarrow k=17$