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The equation of the common tangent touch...

The equation of the common tangent touching the circle `(x-3)^2+y^2=9` and the parabola `y^2=4x` above the x-axis is (a)`sqrt(3)y=3x+1` (b) `sqrt(3)y=-(x+3)` (C)`sqrt(3)y=x+3` (d) `sqrt(3)y=-(3x-1)`

A

`sqrt(3) y = 3x + 1`

B

`sqrt(3) y = - (x + 3)`

C

`sqrt(3) y = x + 3`

D

`sqrt(3) y = - (3x + 1)`

Text Solution

Verified by Experts

The correct Answer is:
C
Any tengent to `y^(2) = 4x` is of the form `y = mx + (1)/(m)`
(`:' a =1`) and this touches the circle `(x - 3)^(2) + y^(2) = 9`
If `|(m(33) + (1)/(m) - 0)/(sqrt(m^(2) + 1))| = 3`
[`:'` centre of the circle is (3,0) and radisu is 3]
`implies (3m^(2) + 1)/(m) = +- 3 sqrt(m^(2) + 1)`
`implies 3m^(2) + 1 = +- 3m sqrt(m^(2) + 1)`
`implies 9m^(4) + 1 + 6m^(2) = 9m^(2) (m^(2) + 1)`
`implies 9m^(4) + 1 + 6m^(2) = 9m^(4) + 9m^(2)`
`implies 3m^(2) = 1`
`implies m = +- (1)/(sqrt(3))`
If the tangent touches the parabola and circle above the X-axis, then slope m should be positive.
`:. m = (1)/(sqrt(3))` and the equation is `y = (1)/(sqrt(3)) x + sqrt(3)`
or `sqrt(3) y = x + 3`
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