Question

The line 2x - y + 6 = 0 meets the circle x2 + y2 -
2y - 9 = 0 at A and B. Find the equation of the
circle on AB as diameter.​

Answer 1

Required circle with AB as diameter is

5x² + 5y² - 28x + 4y + 39 = 0

Step-by-step explanation:

The given equations are

2x - y + 6 = 0 ..... (1)

x² + y² - 2y - 9 = 0 ..... (2)

To find the intersections of the line (2) on circle (1), we substitute either the value of x or y from (1) to (2) no. equation.

From (1), we get

y = 2x - 6 ..... (3)

From (2), we get

x² + (2x - 6)² - 2 (2x - 6) - 9 = 0

or, x² + 4x² - 24x + 36 - 4x + 12 - 9 = 0

or, 5x² - 28x + 39 = 0

or, 5x² - 15x - 13x + 39 = 0

or, 5x (x - 3) - 13 (x - 3) = 0

or, (x - 3) (5x - 13) = 0

So x = 3, 13/5

Putting x = 3 in (3), we get

y = 2 (3) - 6 = 0

and x = 13/5 in (3), we get

y = 2 (13/5) - 6 = - 4/5

Then the two intersections are A (3, 0) and B (13/5, - 4/5) (say)

Therefore the equation of the circle having AB as diameter be

(x - 3) (x - 13/5) + (y - 0) (y + 4/5) = 0

or, x² - (13/5 + 3)x + 39/5 + y² + (4/5)y = 0

or, x² + y² - (28/5)x + (4/5)y + 39/5 = 0

or, 5x² + 5y² - 28x + 4y + 39 = 0