Let A(2,3), B(3,-6), C(5,-7) be three points. If P is a point satisfying the condition P A^2+P B^2=2 P C^2 , then a point that lies on the locus of P is

Step-by-step solution

Given, points are A(2,3), B(3,-6), C(5,-7) . Let point P be (x, y) , then according to condition aligned & P A^2+P B^2=2 P C^2 & aligned (x-2)^2+(y-3)^2+(x-3)^2+(y+6)^2 =2 [(x-5)^2+(y+7)^2 ] aligned & aligned x^2+4-4 x+y^2+9-6 y+x^2+9-6 x +y^2+36+12 y aligned & =2 [x^2+25-10 x+y^2+14 y+49 ] & 2 x^2+2 y^2-10 x+6 y+58 & =2 x^2+2 y^2-20 x+28 y+148 & 10 x-22 y=90 aligned By checking options, 1. 10(2) − 22(− 5) = 20 + 110 =130 2. 10(− 2) − 22(5) = − 20 −110 = −130 3. 10(13) − 22(10) =130 − 220 = − 90 4. 10(−13) − 22(−10