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S1 : If x=3y=2 are the equations of asymptotes of a hyperbola and hyperbola passes through the point (4, 6) then length of its latus rectum is 4\sqrt{2} .

S2 : Two concentric rectangular hyperbolas whose axes meet at an angle \pi/4, cut each other at an angle \pi/2.

S3 : Distance between directrices of hyperbola xy = 16 is 4

S4 :  If line joining the points A(x_{1}0)B(0,y_{1}) is tangent to the hyperbola xy=c^{2} then point of contact is \left ( \frac{x_{1}}{2}, \frac{y_{1}}{2}\right ).

Option: 1

TTFT


Option: 2

TFTT


Option: 3

FFTT


Option: 4

FFTF


Answers (1)

best_answer

 

Equation of Tangent to Hyperbola -

\frac{xx_{1}}{a^{2}}-\frac{yy_{1}}{b^{2}}= 1

- wherein

For the Hyperbola

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

P\left ( x_{1} ,y_{1}\right )

 

 

Rectangular Hyperbola -

x^{2}-y^{2}= a^{2}

- wherein

 

 

 

S1 : Equation of hyperbola (x-3)(y-2)=c^{2}

xy-2x-3y+6=c^{2}

\therefore It passes through (4,6), then

4\times 6-2\times 4-3\times 6+6=c^{2}

c^{2}=4

c=2

Latus rectum (\ l)=2\sqrt{2}\: c=2\sqrt{2}\times 2=4\sqrt{2}

S2 : Let the equation to the rectangular hyperbola be        x^{2}-y^{2}=a^{2}\; \; \; \; \; \; \; \; \; .........(i)

As the asymptotes of this are the axes of the other and vice-versa, hence the equation of the other hyperbola may be written asxy=c^{2}\; \; \; \; \; \; \; ...........(ii)

Let (i) and (ii) meet at some point whose coordinates are (a\; \sec \alpha,a\; \tan \alpha ).

then the tangent at the point (a\; \sec \alpha,a\; \tan \alpha )to equation on (i) is x-y\: \sin \alpha =a\: \cos \alpha\; \; \; \; \; \; \; \; ............(iii)

and the tangent at the point (a\; \sec \alpha,a\; \tan \alpha ) to equation on (ii) is y+x\: \sin \alpha =\frac{2c^{2}}{a}\; \cos \alpha \; \; \; \; \; .........(iv)

So, the slopes of the tangents given by (iii) and (iv) are respectively \frac{1}{\sin \alpha } and -\sin \alpha and their product is -\sin \alpha \times \frac{1}{\sin \alpha }=-1

Hence the tangents are right angle.

S3 : Hyperbola xy=16

\Rightarrow c=4

equation of directrices

x+y=\pm \sqrt{2}c

x+y=\pm 4\sqrt{2}

distance between directrices of hyperbola is \Rightarrow \left | \frac{8\sqrt{2}}{\sqrt{1^{2}+1^{2}}} \right |\Rightarrow \left | \frac{8\sqrt{2}}{\sqrt{2}} \right |=8

S4 : Let point (h,k) on the parabola. then equation of tangent is \frac{x}{h}+\frac{y}{k}=2\; \; \; \; \; \; \; \; \; \; ...........(i)

Equation of line \frac{x}{x_{1}}+\frac{y}{y_{1}}=1

\therefore h=\frac{x_{1}}{2} and k=\frac{y_{1}}{2}

\therefore point of contact is \left ( \frac{x_{1}}{2},\frac{y_{1}}{2} \right )    

Posted by

rishi.raj

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