The point of intersection of the tangents drawn to the hyperbola x^ 2 a^ 2 - y^ 2 b^ 2 = 1 at the points where it is intersected by the line lx + my + n = 0 is&

Correct answer: C. (- a^ 2 l n , b^ 2 m n ) Let,& ( x _ 1 , y _ 1 ) be the required point. Then, the equation of the chord of contact of the tangents drawn from ( x _ 1 , y _ 1 ) to the given hyperbola is x x _ 1 a^ 2 - y y _ 1 b^ 2 = 1 …(i) The given line is lx + my + n = 0 …(ii) Equations (i) and (ii) represent the same line ∴ x _ 1 a^ 2 l = - y _ 1 b^ 2 m = (1)/(- n) ⇒ x _ 1 = - a^ 2 l n , y _ 1 = b^ 2 m n . Hence, the required point is (- a^ 2 l n , b^ 2 m n )

NTA Abhyas 2020

Exam: NTA Abhyas
Year: 2020
Subject: Mathematics
Chapter: Hyperbola

NTA Abhyas 2020 · Mathematics · Hyperbola

The point of intersection of the tangents drawn to the hyperbola x^...

The point of intersection of the tangents drawn to the hyperbola

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

at the points where it is intersected by the line

lx + my + n = 0

is&

Options

A $\left(\dfrac{a^{2}l}{n}, \dfrac{b^{2}m}{n}\right)$
B $\left(\dfrac{a^{2}l}{n}, - \dfrac{b^{2}m}{n}\right)$
C $\left(- \dfrac{a^{2}l}{n}, \dfrac{b^{2}m}{n}\right)$
D $\left(- \dfrac{a^{2}l}{n}, - \dfrac{b^{2}m}{n}\right)$

Correct answer

C. $\left(- \dfrac{a^{2}l}{n}, \dfrac{b^{2}m}{n}\right)$

Step-by-step solution

Let,&

\left({x}_{1}, {y}_{1}\right)

be the required point. Then, the equation of the chord of contact of the tangents drawn from

\left({x}_{1}, {y}_{1}\right)

to the given hyperbola is

\dfrac{x{x}_{1}}{a^{2}} - \dfrac{y{y}_{1}}{b^{2}} = 1

…(i)
The given line is

lx + my + n = 0

…(ii)
Equations (i) and (ii) represent the same line

∴ \dfrac{{x}_{1}}{a^{2}l} = \dfrac{- {y}_{1}}{b^{2}m} = \dfrac{1}{- n}

⇒ {x}_{1} = \dfrac{- a^{2}l}{n}, {y}_{1} = \dfrac{b^{2}m}{n} .<br />Hence,

the required point is

\left(- \dfrac{a^{2}l}{n}, \dfrac{b^{2}m}{n}\right)

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