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The equation of the circle passing throu...

The equation of the circle passing through (1,2) and the point of intersection of the circles
` x^(2) + y^(2) - 8x - 6y + 21 = 0 " and " x^(2) + y^(2) - 2x - 15 = 0 ` is

A

`x^(2) + y^(2) - 6x - 2y + 9 = 0`

B

`x^(2) + y^(2) + 6x - 2y + 9 = 0`

C

`x^(2) + y^(2) - 6x -4y + 9 = 0`

D

`x^(2) + y^(2) - 6x + 4y + 9 = 0`

Text Solution

Verified by Experts

The correct Answer is:
C
Given circle are, `S = x^(2) + y^(2) - 8x - 6y + 21 = 0` and `S. -= x^(2) + y^(2) - 2x - 15 = 0` point p (1,2)
`implies` Equaton of radical axis is `S. - S^(1) = 0`
`implies -6x - 6y + 36 = 0`
`implies L -= x + y - 6 = 0`
`implies` Equation of circle passing through point of intersection of S and `S^(1)` is,
`implies S + lambda L = 0`
`implies (x^(2) + y^(2) - 8x - 6y + 21) + lambda (x + y - 6) = 0`
Equation (1) passes through p (1,2)
`implies 1 + 4 - 8 - 12 + 21 + lambda (1 + 2 - 6) = 0`
`implies 3 lambda = 6`
`implies lambda = 2`
Equation (1) `implies x^(2) + y^(2) - 8x - 6y + 21 + 2x + 2y - 12 = 0`
`:. x^(2) + y^(2) - 6x - 4y + 9 = 0`, is required circle
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Knowledge Check

  • The equation of the circle passing through (1,2) and the points of intersection of the circles x^2+y^2-8x-6y+21=0 and x^2+y^2-2x-15=0 is

    A
    `x^2+y^2+6x-2y+9=0`
    B
    `x^2+y^2-6x-4y+9=0`
    C
    `x^2+y^2-6x-4y+9=0`
    D
    `x^2+y^2-6x+4y+9=0`

  • The equation of the circle passing through the points of intersection of the circles x^(2) + y^(2)- 3x - 6y + 8 = 0 , x^(2)+ y^(2) - 2x - 4y + 4 = 0 and touching the line 2x+ y = 3 is

    A
    `x^(2) + y^(2) + 2x - 4 = 0 , 15x^(2) + 15y^(2)+ 10x +20y - 48 = 0 `
    B
    `x^(2) + y^(2) - 4x - 8y + 12 = 0 , 5x^(2) + 5y^(2) + 4x + 8y - 36 = 0 `
    C
    `x^(2) + y^(2) - 6x - 4y - 9 = 0 ,9x^(2) + 9x^(2) - 30x - 20y - 25 = 0 `
    D
    `x^(2) + y^(2) - 6x - 4y + 9 = 0 , x^(2) + y^(2) - 10x + 50 y - 25 = 0 `

  • The equation of the circle passing through the point of intersection of the circles x^(2) + y^(2) + 6x + 4y - 12 = 0 , x^(2) + y^(2)- 4x - 6y - 12 = 0 and having radius sqrt(13) is

    A
    `9x^(2) + 9y^(2) + 16y - 34 = 0 `
    B
    `10(x^(2) + y^(2)) + 3y 0 86 = 0 `
    C
    `x^(2) +y^(2) - x- 2y = 0 `
    D
    `x^(2) +y^(2) - 2y - 12 = 0 `

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