If line through point $P(a,2)$ meets the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ at A and D and meets the coordinate axis at B and C If line through point $P(a,2)$ meets the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ at $A$ and $D$ and meets the coordinate axes at $B$ and $C$ so that $PA$, $PB$, $PC$, $PD$ are in geometric progression, then the possible values of $a$ can be
$(A)5\hspace{1cm}(B)8\hspace{1cm}(C)10\hspace{1cm}(D)-7$

I could not solve this question, I inferred from question that $PA\cdot PD=PB\cdot PC$ and $PA\cdot PD=PT^2$, where $T$ is the point of tangency.
But I could not solve further. This is a multiple correct choice type question. Please help me.
Thanks.

WLOG we can take $a>0$, $A$ nearer to $P$ than $D$ and notice that the only case we must consider is when $B$ is the intersection with $x$-axis and $C$ is the intersection with $y$-axis, for otherwise those four segments cannot form a geometric progression. If $b$ is the $x$ coordinate of $B$, the equation of line $PB$ is $y=2(x-a)/(a-b)+2$ so that the $y$ coordinate of $C$ is $y_C=-2b/(a-b)$.
Combining this equation with that of the ellipse, we can readily find the $y$ coordinate of $A$ and $D$:
$$
y_A= \frac{2 \left(3 \sqrt{a^2-2 a b+9}-a b+b^2\right)}{a^2-2 a b+b^2+9},
\quad 
y_D= \frac{2 \left(-3 \sqrt{a^2-2 a b+9}-a b+b^2\right)}{a^2-2 a b+b^2+9}. 
$$
We know that $PA:PB=PB:PC=PC:PD$ and this relation also holds for the $y$ components of the segments, that is:
$$
(y_P - y_A):(y_P - y_B) = (y_P - y_B):(y_P - y_C) = (y_P - y_C):(y_P - y_D).
$$
Inserting here the expressions given above for $y_C$, $y_A$, $y_D$, as well as $y_P=2$ and $y_B=0$, we can solve for $a$ and $b$. The only acceptable positive solution is:
$$
a=3 \sqrt{2+\sqrt{13}}\approx 7.10281,
$$
but of course the opposite value, by symmetry, is also a valid solution. As you can see, this is not far from your $(D)$ choice but it is not the same. So the exercise is wrong.

