Variable ellipses are drawn with x=-4 as directrix and the origin as the corresponding focus. The eccentricity of the locus of the extremities of the minor axes of these ellipses is

Question

Variable ellipses are drawn with x=-4 as directrix and the origin as the corresponding focus. The eccentricity of the locus of the extremities of the minor axes of these ellipses is (A) 1 (B) 2 (C) √2 (D) (1/2)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="Solution" src="https://cdn.tardigrade.in/q/nta/m-x6er0tmocqy3.png" /> Now k=b and h=ae Also (a/e)-ae=4⇒ a(1 - e2)=4e ⇒ a2(1 - e2)=4ae ⇒ b2=4h ⇒ k2=4h⇒ y2=4x ⇒ eccentricity =1

Q. Variable ellipses are drawn with $x = - 4$ as directrix and the origin as the corresponding focus. The eccentricity of the locus of the extremities of the minor axes of these ellipses is

$1$

$2$

$2$

$\frac{1}{2}$

Now $k = b$ and $h = a e$
Also $\frac{a}{e} - a e = 4 \Rightarrow a (1 - e^{2}) = 4 e$
$\Rightarrow a^{2} (1 - e^{2}) = 4 a e$
$\Rightarrow b^{2} = 4 h$
$\Rightarrow k^{2} = 4 h \Rightarrow y^{2} = 4 x$
$\Rightarrow$ eccentricity $= 1$

Q. Variable ellipses are drawn with x=−4 as directrix and the origin as the corresponding focus. The eccentricity of the locus of the extremities of the minor axes of these ellipses is

1

2

2​

21​

Now k=b and h=ae Also ea​−ae=4⇒a(1−e2)=4e ⇒a2(1−e2)=4ae ⇒b2=4h ⇒k2=4h⇒y2=4x ⇒ eccentricity =1

Q. Variable ellipses are drawn with $x = - 4$ as directrix and the origin as the corresponding focus. The eccentricity of the locus of the extremities of the minor axes of these ellipses is

$1$

$2$

$2$

$\frac{1}{2}$

Now $k = b$ and $h = a e$
Also $\frac{a}{e} - a e = 4 \Rightarrow a (1 - e^{2}) = 4 e$
$\Rightarrow a^{2} (1 - e^{2}) = 4 a e$
$\Rightarrow b^{2} = 4 h$
$\Rightarrow k^{2} = 4 h \Rightarrow y^{2} = 4 x$
$\Rightarrow$ eccentricity $= 1$