SATHEE: Parabola 2 Question 14

Parabola 2 Question 14

14. Given A circle, $2 x^{2} + 2 y^{2} = 5$ and a parabola, $y^{2} = 4 \sqrt{5} x$ .

Statement I An equation of a common tangent to these curves is $y = x + \sqrt{5}$ .

Statement II If the line, $y = m x + \frac{\sqrt{5}}{m} (m \neq 0)$ is the common tangent, then $m$ satisfies $m^{4} - 3 m^{2} + 2 = 0$ .

(2013 Main)

(a) Statement I is correct, Statement II is correct, Statement II is a correct explanation for Statement I

(b) Statement I is correct, Statement II is correct, Statement II is not a correct explanation for Statement I

(d) Statement I is incorrect, Statement II is correct

Objective Questions II

(One or more than correct option)

Show Answer

Solution:

Equation of circle can be rewritten as $x^{2} + y^{2} = \frac{5}{2}$ .

$Centre \to (0, 0) and radius \to \sqrt{\frac{5}{2}}$

Let common tangent be

$y = m x + \frac{\sqrt{5}}{m} \Rightarrow m^{2} x - m y + \sqrt{5} = 0$

The perpendicular from centre to the tangent is equal to radius of the circle.

$\begin{array}{ll} ∴ & \frac{\sqrt{5} / m}{\sqrt{1 + m^{2}}} = \sqrt{\frac{5}{2}} \\ \Rightarrow & m \sqrt{1 + m^{2}} = \sqrt{2} \\ \Rightarrow & m^{2} (1 + m^{2}) = 2 \\ \Rightarrow & m^{4} + m^{2} - 2 = 0 \\ \Rightarrow & (m^{2} + 2) (m^{2} - 1) = 0 \\ \Rightarrow & m = \pm 1 [∵ m^{2} + 2 \neq 0, as m \in R] \end{array}$

$∴ y = \pm (x + \sqrt{5})$ , both statements are correct as $m \pm 1$ satisfies the given equation of Statement II.

Parabola 2 Question 14

14. Given A circle, $2 x^{2} + 2 y^{2} = 5$ and a parabola, $y^{2} = 4 \sqrt{5} x$ .

Objective Questions II

Solution:

Engineering Preparation

Medical Preparation

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Sample Mock Test

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NCERT Exercise Video Solution

Parabola 2 Question 14

14. Given A circle, 2x2+2y2=5 and a parabola, y2=45x.

Objective Questions II

Solution:

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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14. Given A circle, $2 x^{2} + 2 y^{2} = 5$ and a parabola, $y^{2} = 4 \sqrt{5} x$ .