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Equation of a common tangent to the circle, $x^2 + y^2 - 6x = 0$ and the parabola, $y^2 = 4x$ , is:
Option 1)
$2\sqrt{3}y = 12x +1$
Option 2)
$\sqrt3 y = x +3$
Option 3)
$2\sqrt{3}y = -x -12$
Option 4)
$\sqrt3y = 3x +1$

Answers (1)

best_answer

Angle of intersection of two curves -

The angle of intersection of two curves is the angle subtended between the tangents at their point of intersection.Let m₁ & m₂ are two slope of tangents at intersection point of two curves then

$tan\theta=\frac{[m_{1}-m_{2}]}{1+m_{1}m_{2}}$

- wherein

where $\theta$ is angle between two curves tangents.

From the concept

Equation of tangent to parabola $y^{2}=4ax$

is $y=mx+\frac{a}{m}$

Here, parabola, $y^{2}=4x$

So, tangent $\Rightarrow y=mx+\frac{1}{m}$

This is also a common tangent to circle $x^{2}+y^{2}-6x=0$

$radius =3$

$\Rightarrow \frac{\left | 3m^{2}+1 \right |}{\sqrt{m^{4}+m^{2}}}=3$

$\Rightarrow m=\pm \frac{1}{\sqrt{3}}$

$\Rightarrow$ tangent are $x+\sqrt{3}y+3=0$

and $x-\sqrt{3}y+3=0$

Option 1)

$2\sqrt{3}y = 12x +1$

Option 2)
$\sqrt3 y = x +3$
Option 3)

$2\sqrt{3}y = -x -12$

Option 4)

$\sqrt3y = 3x +1$

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