SATHEE: Parabola 2 Question 5

Parabola 2 Question 5

5. Equation of a common tangent to the circle, $x^{2} + y^{2} - 6 x = 0$ and the parabola, $y^{2} = 4 x$ , is

(2019 Main, 9 Jan, I)

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

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

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

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Answer:

Correct Answer: 5. (d)

Solution:

We know that, equation of tangent to parabola $y^{2} = 4 a x$ is

$y = m x + \frac{a}{m}$

$∴$ Equation of tangent to the parabola $y^{2} = 4 x$ is

$\begin{aligned} y = m x + \frac{1}{m} \\ \Rightarrow & m^{2} x - m y + 1 = 0 \end{aligned}$

Now, let line (i) is also a tangent to the circle.

Equation of circle $x^{2} + y^{2} - 6 x = 0$

Clearly, centre of given circle is $(3, 0)$ and radius $= 3$

$[∵$ for the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ , centre $=$

$(- g, - f) and radius = \sqrt{g^{2} + f^{2} - c]}$

$∴$ The perpendicular distance of $(3, 0)$ from the line (i) is 3.

$[∵$ Radius is perpendicular to the

$\Rightarrow \frac{| m^{2} \cdot 3 - m \cdot 0 + 1 |}{\sqrt{{(m^{2})}^{2} + (- m)^{2}}} = 3$

tangent of circle]

The length of perpendicular from a point $(x_{1}, y_{1})$ to the line $a x + b y + c = 0$ is $| \frac{a x_{1} + b y_{1} + c}{\sqrt{a^{2} + b^{2}}} |$ .

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

$\Rightarrow 9 m^{4} + 6 m^{2} + 1 = 9 (m^{4} + m^{2})$

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

$∵ lim_{m \to \infty} \frac{3 m^{2} + 1}{\sqrt{m^{4} + m^{2}}} = lim_{m \to \infty} \frac{3 + \frac{1}{m^{2}}}{\sqrt{1 + \frac{1}{m^{2}}}} = 3$

$∴$ Equation of common tangents are $x = 0$ ,

$y = \frac{x}{\sqrt{3}} + \sqrt{3} and y = \frac{- x}{\sqrt{3}} - \sqrt{3} using y = m x + \frac{1}{m}$

i.e. $x = 0, \sqrt{3} y = x + 3$ and $\sqrt{3} y = - x - 3$

Parabola 2 Question 5

5. Equation of a common tangent to the circle, $x^{2} + y^{2} - 6 x = 0$ and the parabola, $y^{2} = 4 x$ , is

Answer:

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Parabola 2 Question 5

5. Equation of a common tangent to the circle, x2+y2−6x=0 and the parabola, y2=4x, is

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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5. Equation of a common tangent to the circle, $x^{2} + y^{2} - 6 x = 0$ and the parabola, $y^{2} = 4 x$ , is