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The locus of the centre of the circle, w...

The locus of the centre of the circle, which cuts the circle `x^(2) + y^(2) - 20x + 4 = 0` orthogonally and touchs the line x = 2 , is

A

`x^(2) = 16y`

B

`y^(2) = 4x`

C

`y^(2) = 16 x`

D

`x^(2) = 4y`

Text Solution

Verified by Experts

The correct Answer is:
C
Given circle is , `S = x^(2) + y^(2) - 20 x + 4 = 0` and
Line `x = 2 implies x - 2 = 0`
Let the required circle is , `S^(1) -= x^(2) + y^(2) + 2gx + 2gy + c = 0`
S = 0 cur `S^(1) = 0` orthogonally
`implies - 20 g + 0 = 4 + c [ :. 2gg, + 2ff. = c + c.]`
`implies c = - 20 g - 4`
S = 0 touch the line x = 2
`implies |g + 2| = sqrt(g^(2) + f^(2) - c) [:. c = - 20 g - 4]`
`implies g + 2 = sqrt(g^(2) + f^(2) + 20g + 4)`
`implies g^(2) + 4g = 4 = g^(2) + f^(2) + 20g + 4`
`impies f^(2) + 16 g = 0`
The locus of the centre (-g, f) is `y^(2) - 16x = 0`
`:. y^(2) = 16 x`
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Knowledge Check

  • The locus of the centre of the circle which cuts the circle x^(2) + y^(2) - 20x + 4 = 0 orthogonally and touches the line x = 2 is

    A
    `y^(2) = 4x `
    B
    `y^(2) = 16x `
    C
    `x^(2) = 4y`
    D
    `x^(2) = 16y `

  • The locus of centres of the circles which cut the circles x^2+y^2+4x-6y+9=0, x^2+y^2-5x+4y+2=0 orthogonally is

    A
    3x+4y-5=0
    B
    9x-10y+7=0
    C
    9x+10y-7=0
    D
    9x-10y+11=0

  • The locus of the centre of the circle which cuts the circles x^(2) + y^(2) + 4x - 6y + 9 = 0 " and " x^(2) + y^(2) - 5x + 4y + 2 = 0 orthogonally is

    A
    `3x + 4y - 5 = 0 `
    B
    ` 9x - 10y + 7 = 0 `
    C
    `9x + 10y - 7 = 0 `
    D
    `9x - 10y + 11 = 0 `

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