Question

Let $S$ be a curved mirror passing through $(3,4)$ having the property that all the light emerging from origin(focus) , after getting reflected from the mirror becomes parallel to $x-$axis. The angle between the tangents drawn from the point $(-2,6)$ is ${90}^{\circ }$. If the circle $(x-4{)}^2+y^2=r^2$ internally touches the curve $S,$ then the value of $r^2$ is

Answer 1

In any parabola , if we send a ray from the focus to the mirror curve ,it will be parallel to axis of the parabola after reflection.
So, $(0,0)$ is the focus ,$X-$axis is the axis
The equation of the parabola be $y^2=4a(x-b)$
It passes through $(3,4)$.
$\Rightarrow 16=4a(3-b)\Rightarrow a=\frac{4}{3-b}$
The locus of the point from which perpendicular tangents are drawn to the parabola is directrix.
Directrix is perpendicular to the axis and passes through $(-2,6)$. So, the equation of the directrix is $x=-2$



Vertex is the midpoint of $A(-2,0)$ and $S(0,0)$
$\Rightarrow b=-1\Rightarrow a=1$
The equation of the parabola is $y^2=4(x+1)$


Solving parabola and circle
$(x-4{)}^2+4(x+1)=r^2\Rightarrow x^2-4x+20-r^2=0$
If both curve touch each other ,then $△=0$
$16-4(20-r^2)=0\Rightarrow 20-r^2=4\Rightarrow r^2=16$