First slide

Ellipse

Question

$If tangents are drawn to the ellipse x^{2} + 2 y^{2} = 2, then the locus of the midpoint of$
$the intercept made by the tangents between the coordinate axes is$

Moderate

A

$\frac{1}{2 x^{2}} + \frac{1}{4 y^{2}} = 1$
B

$\frac{1}{4 x^{2}} + \frac{1}{2 y^{2}} = 1$
C

$\frac{x^{2}}{2} + \frac{y^{2}}{4} = 1$
D

$\frac{x^{2}}{4} + \frac{y^{2}}{2} = 1$

Solution

Tangent to ellipse
$\frac{x^{2}}{2} + \frac{y^{2}}{1} = 1 at P (\sqrt{2} \cos θ, \sin θ)$
$is given by$
$\frac{x \cos θ}{\sqrt{2}} + y \sin θ = 1$
$It meets axes at A and B . Let Q(h, k) be midpoint of AB. Using midpoint formula, we have$
$\begin{array}{l} A \equiv (\sqrt{2} \sec θ, 0) and B \equiv (0, cosec θ) \\ Hence, 2 h = \sqrt{2} \sec θ and 2 k = cosec θ \end{array}$
$\begin{array}{l} or {(\frac{1}{\sqrt{2} h})}^{2} + {(\frac{1}{2 k})}^{2} = 1 \\ or \frac{1}{2 h^{2}} + \frac{1}{4 k^{2}} = 1 \\ or \frac{1}{2 x^{2}} + \frac{1}{4 y^{2}} = 1 \end{array}$

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