First slide

Parabola

Question

$The equation of the common tangent touching the circle (x - 3)^{2} + y^{2} = 9 and the parabola y^{2} = 4 x above the x- axis is$

Moderate

A

$\sqrt{3} y = 3 x + 1$
B

$\sqrt{3} y = - (x + 3)$
C

$\sqrt{3} y = x + 3$
D

$\sqrt{3} y = - (3 x + 1)$

Solution

The equation of tangent to the given parabola having slope m is $y = m x + \frac{1}{m} - - - - (1)$
$The equation of tangent to the given circle having slope m is y = m (x - 3) \pm 3 \sqrt{1 + m^{2}} - - - (2)$
Equations ( 1) and (2) are identical. So,
$\begin{array}{ll} \frac{1}{m} = - 3 m \pm 3 \sqrt{1 + m^{2}} \\ or 1 + 3 m^{2} = \pm 3 m \sqrt{1 + m^{2}} \\ or 1 + 6 m^{2} + 9 m^{4} = 9 (m^{2} + m^{4}) \\ or 3 m^{2} = 1 \\ or m = \pm \frac{1}{\sqrt{3}} \end{array}$
$Hence, the equation of common tangent is \sqrt{3} y = x + 3$
(as tangent is lying above the x-axis).

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