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  • The points of contact of the line  with <img alt="\mathrm{3 x^2-4 y^2=12}" sr

The points of contact of the line \mathrm{y=x-1} with \mathrm{3 x^2-4 y^2=12} is 

Option: 1

(4, 3)


Option: 2

(3, 4)


Option: 3

(4, - 3)


Option: 4

None of these


Answers (1)

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The equation of line and hyperbola are \mathrm{y=x-1 \quad \ldots . \text { (i) and } 3 x^2-4 y^2=12\ \ ..........(ii)}
From \mathrm{ (i) } and \mathrm{ (ii) }, we get \mathrm{3 x^2-4(x-1)^2=12}
\mathrm{\Rightarrow 3 x^2-4\left(x^2-2 x+1\right)=12 \text { or } x^2-8 x+16=0 \Rightarrow x=4}
From \mathrm{(i)}, \mathrm{y=3} so points of contact is \mathrm{(4,3)}
Trick : Points of contact are \mathrm{\left( \pm \frac{a^2 m}{\sqrt{a^2 m^2-b^2}}, \pm \frac{b^2}{\sqrt{a^2 m^2-b^2}}\right)}.
Here \mathrm{a^2=4, \quad b^2=3} and \mathrm{m=1} . So the required points of contact is \mathrm{(4, 3).}

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Anam Khan

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