First slide

Equation of line in Straight lines

Question

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

Moderate

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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