NIMCET 2014 — Mathematics PYQ

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Question

NIMCET 2014 — Mathematics PYQ

NIMCET | Mathematics | 2014

The condition that the line $lx + my + n = 0$ becomes a tangent to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ , is:

Choose the correct answer:

Explanation

Solution

Given equation of the line:

$lx + my + n = 0$

Rewriting the line equation in slope-intercept form ( $y = mx + c$ ):

$my = -lx - n$

$y = \left(-\frac{l}{m}\right)x - \frac{n}{m}$

Comparing this with $y = Mx + C$ , we get:

$M = -\frac{l}{m}$

$C = -\frac{n}{m}$

For a line to be tangent to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ , the condition is:

$C^{2} = a^{2}M^{2} + b^{2}$

Substituting the values of $M$ and $C$ :

$\left(-\frac{n}{m}\right)^{2} = a^{2}\left(-\frac{l}{m}\right)^{2} + b^{2}$

$\frac{n^{2}}{m^{2}} = \frac{a^{2}l^{2}}{m^{2}} + b^{2}$

Multiplying the entire equation by $m^{2}$ :

$n^{2} = a^{2}l^{2} + b^{2}m^{2}$

Correct Option: 4