First slide

Ellipse

Question

The locus of the points of intersection of
the tangents at the extremities of the chords of the ellipse
$x^{2} + 2 y^{2} = 6$ which touch the ellipse $x^{2} + 4 y^{2} = 4$ is

Moderate

A

$x^{2} + y^{2} = 4$
B

$x^{2} + y^{2} = 6$
C

$x^{2} + y^{2} = 9$
D

none of these

Solution

We can write $x^{2} + 4 y^{2} = 4 as \frac{x^{2}}{4} + \frac{y^{2}}{1} = 1 (i)$
Equation of a tangent to the ellipse $(i)$ is
   $\frac{x}{2} \cos θ + y \sin θ = 1$
Equation of the ellipse $x^{2} + 2 y^{2} = 6$ can be written as
   $\frac{x^{2}}{6} + \frac{y^{2}}{3} = 1$
Suppose $(i i)$ meets the ellipse $(i i i)$ at P and Q and the tan
gents at $P$ and $Q$ to the ellipse $(i i i)$ intersect at $(h, k)$ , then
$(i i)$ is the chord of contact of $(h, k)$ with respect to the ellipse
$(i i i)$ and thus its equation is $\frac{h x}{6} + \frac{k y}{3} = 1$ $(iv)$
Since $(i i)$ and $(i v)$ represent the same line
$\begin{array}{l} \frac{h / 6}{(\cos θ) / 2} = \frac{k / 3}{\sin θ} = 1 \\ \Rightarrow h = 3 \cos θ, k = 3 \sin θ . \end{array}$
and the locus of $(h, k)$ is $x^{2} + y^{2} = 9$

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