Question

The locus of the mid-point of the distance between the axes of the variable line $x\cos\alpha + y\sin\alpha = p,$ where p is constant, is.

• A. $x^{2} + y^{2} = 4p^{2}$
• B. $\frac{1}{x^{2}} + \frac{1}{y^{2}} = \frac{4}{p^{2}}$
• C. $x^{2} + y^{2} = \frac{4}{p^{2}}$
• D. $\frac{1}{x^{2}} + \frac{1}{y^{2}} = \frac{2}{p^{2}}$

Answer 1

Answer: (B)

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

Answer 2

The straight line $x \cos \alpha + y \sin \alpha = p$ meets the x-axis at the point $A \left( \frac { p } { \cos \alpha } , 0 \right)$ and the y-axis at the point $B \left( 0 , \frac { p } { \sin \alpha } \right)$. Let (h, k) be the coordinates of the middle point of the line segment AB.

Then $h = \frac { p } { 2 \cos \alpha }$ and $k = \frac { p } { 2 \sin \alpha }$

$\Rightarrow \cos \alpha = \frac { p } { 2 h }$ and $\sin \alpha = \frac { p } { 2 k }$

$\Rightarrow \sin ^ { 2 } \alpha + \cos ^ { 2 } \alpha = \frac { p ^ { 2 } } { 4 h ^ { 2 } } + \frac { p ^ { 2 } } { 4 k ^ { 2 } } = 1$

Hence locus of the point (h, k) is $\frac { 1 } { x ^ { 2 } } + \frac { 1 } { y ^ { 2 } } = \frac { 4 } { p ^ { 2 } }$.