The point of concurrence of all conjugate lines of the line 5 x+7 y-78=0 with respect to the circle x^2+y^2+6 x+8 y-96=0 is

Step-by-step solution

The all conjugate lines of the line 5 x+7 y-78=0 with respect to circle x^2+y^2+6 x+8 y-96=0 passes through the pole of the given line with respect to given circle. Let required pole is P (x₁, y₁ ) , so equation of polar of point (P (x₁, y₁ ) . with respect to given circle is T=0 . x x₁+y y₁+3 (x+x₁ )+4 (y+y₁ )-96=0 The line (i) represent the line 5 x+7 y-78=0 only. So, x₁+3 5 = y₁+4 7 = 3 x₁+4 y₁-96 -78 =k (Let) x₁=5 k-3, y₁=7 k-4 and 3 x₁+4 y₁=96-78 k So, 3(5 k-3)+4(7 k-4)=96-78 k array ll & 15 k+28 k+78 k=96+9+1