The number of values of m for which the line y - mx - -m2 - 4 touches the hyperbola 4 x2 - 1 - y2 is -

The correct option is C infiniteEquation of hyperbola may be written as, x^21 y^24=1 ⇒ a^2=1, b^2=4 Now using formula for tangency of a line to a hype ...

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Byju's Answer

Standard XII

Mathematics

Particular Solution of a Differential Equation

The number of...

Question

The number of values of $m$ for which the line $y = m x + \sqrt{m^{2} - 4}$ touches the hyperbola $4 (x^{2} - 1) = y^{2}$ is :

two

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zero

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one

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infinite

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Solution

The correct option is C infinite
Equation of hyperbola may be written as,
$\frac{x^{2}}{1} - \frac{y^{2}}{4} = 1$
$\Rightarrow a^{2} = 1, b^{2} = 4$
Now using formula for tangency of a line to a hyperbola $c^{2} = a^{2} m^{2} - b^{2}$
$\Rightarrow m^{2} - 4 = 1 (m^{2}) - 4 = m^{2} - 4$
$\Rightarrow 0 = 0$
So given equation is tangent to the hyperbola for any value of $m$
Hence, option 'D' is correct.

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The number of values of m for which the line y=mx+√m2−4 touches the hyperbola 4(x2−1)=y2 is :

The correct option is C infiniteEquation of hyperbola may be written as,x21−y24=1⇒a2=1,b2=4Now using formula for tangency of a line to a hyperbola c2=a2m2−b2⇒m2−4=1(m2)−4=m2−4⇒0=0So given equation is tangent to the hyperbola for any value of mHence, option 'D' is correct.

The number of values of $m$ for which the line $y = m x + \sqrt{m^{2} - 4}$ touches the hyperbola $4 (x^{2} - 1) = y^{2}$ is :