First slide

Tangents and normals

Question

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

$Equation of the tangent to y^{2} = 4 x is y = m x + \frac{1}{m} \to (1)$
(1) is a tangent to circle with center ( 3, 0) and r = 3

$\begin{array}{l} r = d \Rightarrow 3 = \frac{3 m + \frac{1}{m}}{\sqrt{1 + m^{2}}} \\ m = \pm \frac{1}{\sqrt{3}} \\ I f m = \frac{1}{\sqrt{3}} \Rightarrow y = \frac{x}{\sqrt{3}} + \sqrt{3} \end{array}$

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