Question

Let an ellipse having major axis and minor axis parallel to the $x-$axis and the $y-$axis respectively. Its two foci $S$ and $S^{\prime }$ are $(3,4)$, $(5,4)$ and the line $x-3y+17=0$ is a tangent to the ellipse at a point $P.$ Then

• A. Eccentricity of the ellipse is $\frac{1}{3}$
• B. Length of semi-minor axis is $2\sqrt{2}$
• C. The radius of the director circle of the ellipse is $4$
• D. The equation of the one of the directrices is $x=9$

Answer 1

Answer: (A), (B)

The correct options are
A Eccentricity of the ellipse is $\frac{1}{3}$
B Length of semi-minor axis is $2\sqrt{2}$

The centre of the ellipse is the midpoint of $S{S}^{\prime }\Rightarrow (4,4)$
The equation of the ellipse is
$\frac{(x-4{)}^2}{a^2}+\frac{(y-4{)}^2}{b^2}=1$
Where $a$ and $b$ are the semi-major axis and $b$ is the semi-minor axis, respectively.
$S{S}^{\prime }=2ae\Rightarrow ae=\mathrm{1......}\left(1\right)$

Any tangent to the given ellipse (in slope form)
$(y-4)=m(x-4)\pm \sqrt{a^2{m}^2+b^2}\Rightarrow y=mx+4-4m\pm \sqrt{a^2{m}^2+b^2}$
Comparing it with the equation $y=\frac{x}{3}+\frac{17}{3}$

$m=\frac{1}{3},4-4m\pm \sqrt{a^2{m}^2+b^2}=\frac{17}{3}\Rightarrow a^2+9b^2=\mathrm{81....}\left(2\right)$
From equation $\left(1\right)$ and $\left(2\right)$
$a^2=9\&b^2=8$

The equation of the ellipse is
$\frac{(x-4{)}^2}{9}+\frac{(y-4{)}^2}{8}=1$

The equation of the director circle is $(x-4{)}^2+(y-4{)}^2=9+8=17$
The equation of the directrix is $x-4=\pm \frac{a}{e}$
$\Rightarrow x-4=\pm 9\Rightarrow x=13\text{or}x=-5$