Question

The locus of the moving point P, such that 2PA = 3PB where A is (0,0) and B is (4,–3), is.

• A. $5x^{2} - 5y^{2} - 72x + 54y + 225 = 0$
• B. $5x^{2} - 5y^{2} + 72x + 54y + 225 = 0$
• C. $5x^{2} + 5y^{2} + 72x + 54y + 225 = 0$
• D. $5x^{2} + 5y^{2} - 72x + 54y + 225 = 0$

Answer 1

Answer: (D)

$5x^{2} + 5y^{2} - 72x + 54y + 225 = 0$

Answer 2

Let $P ( h , k )$ Given $2 P A = 3 P B \Rightarrow 4 P A ^ { 2 } = 9 P B ^ { 2 }$

$\Rightarrow 4 \left( h ^ { 2 } + k ^ { 2 } \right) = 9 \left[ ( h - 4 ) ^ { 2 } + ( k + 3 ) ^ { 2 } \right]$

$\Rightarrow 5 h ^ { 2 } + 5 k ^ { 2 } - 72 h + 54 k + 225 = 0$

Hence the locus of point P is given by

$5 x ^ { 2 } + 5 y ^ { 2 } - 72 x + 54 y + 225 = 0$.