Question

Which among the following statements are CORRECT for an ellipse in the $xy$ plane whose focii are at $(9,20)$and $(49,55)$ and having $x$ axis as a tangent to it

• A. Length of major axis is $85$ unit
• B. Length of minor axis is $20\sqrt{11}$ unit
• C. Co-ordiante of the point where $x$ axis touches the ellipse are $(97,0)$
• D. $(4,8)$ lies inside the ellipse

Answer 1

Answer: (A), (B), (D)

The correct options are
A Length of major axis is $85$ unit
B Length of minor axis is $20\sqrt{11}$ unit
D $(4,8)$ lies inside the ellipse

Let $F_1\equiv (9,20)$ and $F_2=(49,55)$
$x-$ axis is the tangent to the ellipse, it is an angular bisector.
So image of $F_2$ with respect to $x-$ axis lies on $F_{1}P$

Image of $F_1(49,55)$ with respect to $x-$ axis is $F_2^{\prime }(49,-55)$

So, equation of $F_{1}P$ is
$y-20=\frac{-55-20}{49-9}(x-9)$
$y-20=\frac{-15}{8}(x-9)$
So point $P$ is $(\frac{59}{3},0)$

Product of length of perpendicular from foci upon any tangent
$=b^2$
$\left(F_{1}A\right)\left(F_{1}B\right)=b^2$
$b^2=20\times 55$
$b=10\sqrt{11}$ unit

$F_1{F}_2=2ae=\sqrt{2825}$
$4(a^2-b^2)=2825a^2=\frac{2825}{4}+1100a=\sqrt{\frac{7225}{4}}=\frac{85}{2}$ unit

So, length of major axis $=85$ unit

For $P(4,8)$
$P{F}_1+P{F}_2<2a$
So, $P$ lies inside ellipse.