Question

The coordinates of the point A and B are $(ak,0)$ and $\left( \frac{a}{k},0 \right),(k = \pm 1)$. If a point P moves so that $PA = kPB,$ then the equation to the locus of P is.

• A. $k^{2}(x^{2} + y^{2}) - a^{2} = 0$
• B. $x^{2} + y^{2} - k^{2}a^{2} = 0$
• C. $x^{2} + y^{2} + a^{2} = 0$
• D. $x^{2} + y^{2} - a^{2} = 0$

Answer 1

Answer: (D)

$x^{2} + y^{2} - a^{2} = 0$

Answer 2

$( x - a k ) ^ { 2 } + y ^ { 2 } = k ^ { 2 } \left[ \left( x - \frac { a } { k } \right) ^ { 2 } + y ^ { 2 } \right]$

$\Rightarrow \left( 1 - k ^ { 2 } \right) \left( x ^ { 2 } + y ^ { 2 } \right) - 2 a k x + 2 a k x + a ^ { 2 } k ^ { 2 } - a ^ { 2 } = 0$

$\Rightarrow x ^ { 2 } + y ^ { 2 } - a ^ { 2 } = 0$.