NIMCET 2020 — Mathematics PYQ

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Question

NIMCET 2020 — Mathematics PYQ

NIMCET | Mathematics | 2020

The tangent to an ellipse $x^{2} + 16 y^{2} = 16$ and making angle 60° with x-axis is:

Choose the correct answer:

Explanation

Solution

1. Standard Form of Ellipse:

Divide the equation by $16$ :

\frac{x^2}{16} + \frac{16y^2}{16} = \frac{16}{16}

\frac{x^2}{16} + \frac{y^2}{1) = 1

Comparing with $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ :

$a^2 = 16$
$b^2 = 1$

2. Finding the Slope ( $m$ ):

The tangent makes an angle $\theta = 60^\circ$ with the $x$ -axis.

m = \tan(60^\circ) = \sqrt{3}

3. Equation of Tangent:

The equation of a tangent to an ellipse with slope $m$ is:

y = mx \pm \sqrt{a^2m^2 + b^2}

4. Substituting the values:

y = \sqrt{3}x \pm \sqrt{16(\sqrt{3})^2 + 1}

y = \sqrt{3}x \pm \sqrt{16(3) + 1}

y = \sqrt{3}x \pm \sqrt{48 + 1}

y = \sqrt{3}x \pm \sqrt{49}

y = \sqrt{3}x \pm 7

Final Answer:

The equations of the tangents are:

\sqrt{3}x - y \pm 7 = 0