NIMCET 2015 — Mathematics PYQ

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Question

NIMCET 2015 — Mathematics PYQ

NIMCET | Mathematics | 2015

If $3x + 4y + k = 0$ is tangent to the hyperbola $9x^2 - 16y^2 = 144$ , then value of $k$ is:

Choose the correct answer:

Explanation

Standard form of the hyperbola:

\frac{9x^2}{144} - \frac{16y^2}{144} = 1

\frac{x^2}{16} - \frac{y^2}{9} = 1

Here, $a^2 = 16$ and $b^2 = 9$ .

Equation of the line:

4y = -3x - k

y = \left(-\frac{3}{4}\right)x + \left(-\frac{k}{4}\right)

This is in the form $y = mx + c$ , where $m = -\frac{3}{4}$ and $c = -\frac{k}{4}$ .

Condition for tangency to a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ :

c^2 = a^2m^2 - b^2

Substituting the values:

\left(-\frac{k}{4}\right)^2 = 16\left(-\frac{3}{4}\right)^2 - 9

\frac{k^2}{16} = 16\left(\frac{9}{16}\right) - 9

\frac{k^2}{16} = 9 - 9

\frac{k^2}{16} = 0

k^2 = 0

k = 0

Correct Option: (a)