Locus of mid point of the portion between the axes of x+y=p where p is constant is:

Question

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

Accepted Answer

Intercepts: $A=\left(\dfrac p{\cos\alpha},0\right)$, $B=\left(0,\dfrac p{\sin\alpha}\right)$.

Midpoint $(x_1,y_1) = \left(\dfrac p{2\cos\alpha},\dfrac p{2\sin\alpha}\right)$, so $\cos\alpha=\dfrac p{2x_1},\ \sin\alpha=\dfrac p{2y_1}$.

$$\cos^2\alpha+\sin^2\alpha=1 \implies \frac{p^2}{4x_1^2}+\frac{p^2}{4y_1^2}=1 \implies \frac1{x_1^2}+\frac1{y_1^2}=\frac4{p^2}$$

Solutions · 1