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

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Aicharya · Accepted Answer

Equation of $A B$ is $x \cos \alpha+y \sin \alpha=p \Rightarrow \frac{x \cos \alpha}{p}+\frac{y \sin \alpha}{p}=1 \Rightarrow \frac{x}{p / \cos \alpha}+\frac{y}{p / \sin \alpha}=1$ So co-ordinates of $A$ and $B$ are $\left(\frac{p}{\cos \alpha}, 0\right)$ and $\left(0, \frac{p}{\sin \alpha}\right)$; So coordinates of mid point of $A B$ are $\left(\frac{p}{2 \cos \alpha}, \frac{p}{2 \sin \alpha}\right)=\left(x_1, y_1\right)($ let $) ; x_1=\frac{p}{2 \cos \alpha} \& y_1=\frac{p}{2 \sin \alpha}$ $\Rightarrow \cos \alpha=p / 2 x_1$ and $\sin \alpha=p / 2 y_1 ; \cos ^2 \alpha+\sin ^2 \alpha=1 \Rightarrow \frac{p^2}{4}\left(\frac{1}{x_1^2}+\frac{1}{y_1^2}\right)=1$ Locus of $\left(x_1, y_1\right)$ is $\frac{1}{x^2}+\frac{1}{y^2}=\frac{4}{p^2}$.

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

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