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Question
If the lines x + y = 6 and x + 2y = 4 are diameters of the circle, and the circle passes through the point (2, 6) then find its equation.
Solution
To get coordinates of centre we should solve the equations of the diameters x + y = 6, x + 2y = 4.
x + y = 6 ……. (1)
x + 2y = 4 ………. (2)
(1) – (2) ⇒ -y = 2
y = -2
Using y = -2 in (1) we get x – 2 = 6
x = 8
Centre is (8, -2) the circle passes through the point (2, 6).
∴ Radius = `sqrt((8 - 2)^2 + (- 2 - 6)^2)`
`= sqrt(6^2 + (- 8)^2)`
`= sqrt(36 + 64)`
`= sqrt100` = 10
Equation of the circle with centre (h, k) and radius r is (x – h)2 + (y – k)2 = r2
⇒ (x – 8)2 + (y + 2)2 = 102
⇒ x2 + y2 – 16x + 4y + 64 + 4 = 100
⇒ x2 + y2 – 16x + 4y – 32 = 0
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