<s>[INST] Finding the locus - middle point of a line segment Question:
Find the locus of the middle point of the portion of the line $x\cos \alpha + y\sin \alpha = p$ which is intercepted between the axes, given that $p$ remains constant.
No idea how to even approach this problem. Please help!
 [/INST] Locus is the set of points that satisfy the given condition.
Here, let $M(h,k)$ is the general point and the given condition is that it must be the midpoint of the distance between the axes of the line $x\cos{\alpha}+y\sin{\alpha}=p$.
We need to find the equation of the locus in terms of $x-$ and $y-$ axis.
$M(h,k)=(\frac{p}{2\cos\alpha},\frac{p}{2\sin\alpha})$
So,
$$
x^2+y^2=(\frac{p}{2\cos\alpha})^2+(\frac{p}{2\sin\alpha})^2=\frac{p^2}{4.\sin^2\alpha\cos^2\alpha}
$$
For $M(h,k)$ we have $x=\frac{p}{2\cos\alpha}$ and $y=\frac{p}{2\sin\alpha}$$\implies \cos\alpha=\frac{p}{2.x}$ and $\sin\alpha=\frac{p}{2y}$
Substituting,
$$
x^2+y^2=\frac{p^2}{4}.\frac{4.y^2.4.x^2}{p^4}=\frac{4.x^2y^2}{p^2}\implies\frac{1}{x^2}+\frac{1}{y^2}=\frac{4}{p^2}
$$
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