If 2 x - y + 1 = 0 is a tangent to the hyperbola x 2 a 2 - y 2 16 = 1 , then which of the following CANNOT be sides of a right angled triangle?

The line y = m x + c is tangent to hyperbola x 2 a 2 - y 2 b 2 = 1 , , if c 2 = a 2 m 2 - b 2 ∴ 1 2 = 4 a 2 - 16 ⇒ a 2 = 17 4 ⇒ a = 17 2 For 2 a , 4, 1 , sides are 17 , 4 , 1 ( ⇒ Right angled triangle) For 2a, 8, 1 , sides are 17 , 8 , 1 ( ⇒ Triangle is not possible) For a, 4, 1 , sides are 17 2 , 4 , 1 ( ⇒ Triangle is not possible) For a, 4, 2 , sides are 17 2 , 4 , 2 ( ⇒ Triangle exist but not right angled)

Mathematics - Hyperbola Question with Solution | TestHub

MathematicsHyperbolaMiscellaneous/MixedEasy2 minPYQ_2017

MathematicsEasymultiple choice

If $2 x - y + 1 = 0$ is a tangent to the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{16} = 1$ , then which of the following CANNOT be sides of a right angled triangle?

Options:(select one or more)

Answer:

A, B, D

Solution:

The line $y = m x + c$ is tangent to hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1,$ , if $c^{2} = a^{2} m^{2} - b^{2}$

$∴ {(1)}^{2} = 4 a^{2} - 16 \Rightarrow a^{2} = \frac{17}{4}$

$\Rightarrow a = \frac{\sqrt{17}}{2}$

For $2 a, 4, 1$ , sides are $\sqrt{17}, 4, 1$ ( $\Rightarrow$ Right angled triangle)

For $2a, 8, 1$ , sides are $\sqrt{17}, 8, 1$ ( $\Rightarrow$ Triangle is not possible)

For $a, 4, 1$ , sides are $\frac{\sqrt{17}}{2}, 4, 1$ ( $\Rightarrow$ Triangle is not possible)

For $a, 4, 2$ , sides are $\frac{\sqrt{17}}{2}, 4, 2$ ( $\Rightarrow$ Triangle exist but not right angled)

Stream:JEE_ADVSubject:MathematicsTopic:HyperbolaSubtopic:Miscellaneous/Mixed

⏱ 2mℹ️ Source: PYQ_2017

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