First slide

Equation of line in Straight lines

Question

The locus of the mid-point of the portion intercepted between the axes by the line x cos $a + y$ sin $a = p$ , where $p$ is a constant, is

Moderate

A

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

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

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

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

Solution

The line $x \cos α + y \sin α = p$ meets the coordinate axes at $A (\frac{μ}{\cos α}, 0) and B (0, \frac{p}{\sin α}) .$
Let $(h, k)$ be the coordinates of the mid point of the portion $A B$ intercepted between the axes by the line x cos a+ y sin $o d = p .$ Then
$\begin{array}{l} \frac{p}{h} = \frac{\cos α}{2}, k = \frac{0 + \sin α}{2} \\ \Rightarrow \cos α = \frac{p}{2 h}, \sin α = \frac{p}{2 k} \\ \Rightarrow \cos^{2} α + \sin^{2} α = \frac{p^{2}}{4 h^{2}} + \frac{p^{2}}{4 k^{2}} \\ \Rightarrow \frac{p^{2}}{4 h^{2}} + \frac{p^{2}}{4 k^{2}} = 1 \end{array}$
Hence, the locus of $(h, k)$ is
$\frac{p^{2}}{4 x^{2}} + \frac{p^{2}}{4 y^{2}} = 1 or, \frac{1}{x^{2}} + \frac{1}{y^{2}} = \frac{t}{p^{2}}$

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