Question

The line 2x – y + 6 = 0 meets the circle x + y + 10x + 9 = 0 at A and B. Find the equation of circle with AB as diameter.

Answer 1

2x – y + 6 = 0 

⇒ y = 2x + 6 

Substituting y = 2x + 6 in x² + y² + 10x + 9 = 0, we get

⇒ x² + (2x + 6)² + 10x + 9 = 0 

⇒ x² + 4x² + 24x + 36 + 10x + 9 = 0 

⇒ 5x² + 34x + 45 = 0 

⇒ 5x² + 25x + 9x + 45 = 0 

⇒ (5x + 9) (x + 5) = 0 

⇒ 5x = -9 or x = -5

⇒ x = -9/5 or x = -5

When x = -9/5,

y = 2 × -9/5 + 6

= -18/5 + 6

=  -18 + 30/5

= 12/5

∴ Point of intersection is A (-9/5, 12/5)

When x = -5, 

y = -10 + 6 = -4 

∴ Point of intersection in B (-5, -4). 

By diameter form, equation of circle with AB as diameter is

(x + 9/5) (x + 5) + (y - 12/5) (y +4) = 0

⇒ (5x + 9) (x + 5) + (5y – 12) (y + 4) = 0 

⇒ 5x² + 25x + 9x + 45 + 5y² + 20y – 12y – 48 = 0 

⇒ 5x² + 5y² + 34x + 8y – 3 = 0