Question

please answer this question with step by step explanation.​

Answer 1

Answer:

2

Condition if two circles are orthogonal

$2 g_{1}g_{2} + 2f_{1}f_{2} = c_{1} + c_{2}$

Now let equation of third circle be

${x}^{2} + {y}^{2} - 2hx - 2ky + c = 0$

It's Centre is ( h , k )

Use condition to both circles of orthogonal

For first circle

f = -2 , g = 3 , c' = 9

So

$2(h)( - 2) + 3 \times 2k = 9 + c \\ 6k - 4h = 9 + c$

Again for second circle

$2 \times 2h + 2 \times ( - 3)k = 4 + c \\ 4h - 6k = 4 + c$

Subtract both equation to eliminate c

$12k - 8h = 5 \\ 12k - 8h - 5 = 0 \\ 8h - 12k + 5 = 0$

Replace variable (h , k) by (x , y)

$8x - 12y + 5 = 0$