Question

Find the locus of the mid-point of the portion of the line x cos α + y sin α = p which is intercepted between the axes.

 

Answer 1

The given line is \[x\cos\alpha + y\sin\alpha = p\] 
We need to find the intersection of the above line with the coordinate axes.
Let us put x = 0, and y = 0, respectively.
Thus,
at x = 0,

\[x\cos\alpha + y\sin\alpha = p\]
at y = 0,

\[x\cos\alpha + 0 = p \Rightarrow x = psec\alpha\]

So, the points on the axes are

\[x\cos\alpha + 0 = p \Rightarrow x = psec\alpha\]

Let P(h, k) be the mid-point of the line AB.
\[\therefore h = \frac{p\sec\alpha + 0}{2}\text{ and }k = \frac{0 + pcosec\alpha}{2}\]
\[ \Rightarrow \cos\alpha = \frac{p}{2h}\text{ and }\sin\alpha = \frac{p}{2k}\]
We know that

\[\sin^2 \alpha + \cos^2 \alpha = 1\]
\[\therefore \left( \frac{p}{2h} \right)^2 + \left( \frac{p}{2k} \right)^2 = 1\]
\[ \Rightarrow \frac{1}{h^2} + \frac{1}{k^2} = \frac{4}{p^2}\]
Hence, the locus of (h, k) is \[\frac{1}{x^2} + \frac{1}{y^2} = \frac{4}{p^2}\].