Locus of mid point of the portion between the axes o xcos⁡α+ysin⁡α=p where p is constant is

Locus of mid point of the portion between the axes o $x \cos α + y \sin α = p$ where $p$ is constant is

A
$x^{2} + y^{2} = 4 / p^{2}$
B
$x^{2} + y^{2} = 4 p^{2}$
C
$1 / x^{2} + 1 / y^{2} = 2 / p^{2}$
D
$1 / x^{2} + 1 / y^{2} = 4 / p^{2}$

Solution:

If $(h, k)$ is the mid-point, then

$h = p / 2 \cos α, k = p / 2 \sin α$

so $(p / 2 h)^{2} + (p / 2 k)^{2} = \cos^{2} α + \sin^{2} α = 1$

$\Rightarrow 1 / h^{2} + 1 / k^{2} = 4 / p^{2}$

Locus of $(h, k)$ is $1 / x^{2} + 1 / y^{2} = 4 / p^{2}$

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