Find the equation of the circle passing through the point (1, 4)and (5

1.	Find the equation of the circle passing through the point (1, 4)and (5, 2) and has its centre on the line x - 2y + 9 =0.
Answer» Let equation of the required circle is x²+ y² > + 2gy + 2fy + c = 0 Given this equation passes through the points (1,-4) and (5,2) (1,4) (1)² + (-4)² 2g(1) + 2f(-4) + c = 0 2g - 8f + C +17 = 0 ... (1) (5,2) 52 +22 + 2g (5) 2f(2) + c= 0 10g + 4f + C + 29 = 0 .... (2) Also center (-g,-f) lies on the line x - 2 y + 9 = 0 -g -2(-f) + c = 0 -g + 2f + c = 0 ... (3) Solving equations 1,2 and 3 we get G = -3 f = 3,c = -47 G = 1 f = 2, c = 4 The required equation are x² + y² + 6x - y - 47 = 0 x² + y² - 2x - 4y + 4 = 0

Find the equation of the circle passing through the point (1, 4)and (5, 2) and has its centre on the line x - 2y + 9 =0.

Answer»

Let equation of the required circle is

x²+ y² > + 2gy + 2fy + c = 0

Given this equation passes through the points

(1,-4) and (5,2)

(1,4) (1)² + (-4)² 2g(1) + 2f(-4) + c = 0

2g - 8f + C +17 = 0 ... (1)

(5,2) 52 +22 + 2g (5) 2f(2) + c= 0

10g + 4f + C + 29 = 0 .... (2)

Also center (-g,-f) lies on the line x - 2 y + 9 = 0

-g -2(-f) + c = 0

-g + 2f + c = 0 ... (3)

Solving equations 1,2 and 3 we get

G = -3 f = 3,c = -47

G = 1 f = 2, c = 4

The required equation are

x² + y² + 6x - y - 47 = 0

x² + y² - 2x - 4y + 4 = 0

Discussion

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