The equation of the common tangent touching the circle (x – 3)2 + y2 = 9 and the parabola y2 = 4x above the x-axis, is

Engineering

Mathematics

Tangent of Parabola

Question

The equation of the common tangent touching the circle (x – 3)² + y² = 9 and the parabola y² = 4x above the x-axis, is

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

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

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

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

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Any tangent to y² = 4x is y = mx + $\frac{1}{m}$ . It touches the circle, if 3 = $| \frac{3 m + \frac{1}{m}}{\sqrt{1 + m^{2}}} |$ .
or   9(1 + m²) = ${(3 m + \frac{1}{m})}^{2}$
or   $\frac{1}{m^{2}} = 3$ ,   $∴$ m = ± $\frac{1}{\sqrt{3}}$ .
For the common tangent to be above the x-axis, m = $\frac{1}{\sqrt{3}}$
$∴$ Common tangent is,
$y = \frac{1}{\sqrt{3}} x + \sqrt{3}$    ⇒   $\sqrt{3}$ y = x + 3

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