Question

A tangent is drawn to the ellipse $E_1:9x^2+y^2=36$ to cut the ellipse $E_2:3x^2+y^2=48$ at the points $A$ and $B$. If the tangents at $A$ and $B$ to the ellipse $E_2$ intersect at $C(p,3),p>0,$ then the value of $\left[p\right]$ is ($[.]$ represents the greatest integer function)

Answer 1

$E_1:\frac{x^2}{4}+\frac{y^2}{36}=1$
Equation of any tangent $AB$ to the ellipse $E_1$ is $\frac{2x\cos \theta }{4}+\frac{6y\sin \theta }{36}=1\cdots \left(1\right)$

Let tangents at $A$ and $B$ intersect at $C(h,k)$.
Then $AB$ is the chord of contact for $E_2$ w.r.t. point $C$.
So, equation of $AB$ is $T=0$
i.e., $\frac{hx}{16}+\frac{ky}{48}=1\cdots \left(2\right)$

Comparing $\left(1\right)$ and $\left(2\right)$, we get
$\frac{\cos \theta }{2}=\frac{h}{16},\frac{\sin \theta }{6}=\frac{k}{48}$
$\Rightarrow \cos \theta =\frac{h}{8},\sin \theta =\frac{k}{8}$
$\Rightarrow h^2+k^2=64$
$\therefore p^2+3^2=64[\because (h,k)\equiv (p,3\left)\right]$
$\Rightarrow p=\sqrt{55},(\because p>0)$