Test Index

Hyperbola

Section: Mathematics

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Question : 35 of 98

Marks: +1, -0

Let

m_1

m_2

be the slopes of the tangents drawn from the point

P(4,1)

to the hyperbola

H: \;\frac{y^2}{25}-\;\frac{x^2}{16}=1

Q

is the point from which the tangents drawn to

H

\left| m_1 \right|

\left| m_2 \right|

and they make positive intercepts

\alpha

\beta

x

\;\frac{(PQ)^2}{\alpha\beta}

is equal to _______.

[13-Apr-2023 shift 1]

Your Answer:

Solution:

Equation of tangent to the hyperbola

\;\frac{y^2}{a^2}-\;\frac{x^2}{b^2}=1

y=m x \pm \sqrt{a^2-b^2 m^2}

passing through

(4,1)

\;1=4 m \pm \sqrt{25-16 m^2} \Rightarrow 4 m^2 - m -3 =0

\;\Rightarrow m=1, \;\frac{-3}{4}

Equation of tangent with positive slopes

1 \& \;\frac{3}{4}

4 y=3 x-16

y=x-3\}

with positive intercept on

x

-axis.

\alpha=\;\frac{16}{3}, \beta=3

Intersection points:

\;Q:(-4,-7)

\;P:(4,1)

\;P Q^2=128

\;\;\frac{P^2}{\alpha\beta} =\;\frac{128}{16} = 8

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