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The locus of a point P(\alpha ,\beta ) moving under the condition that the line y=\alpha x+\beta is a tangent to the hyperbola \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1   is

  • Option 1)

    a circle

  • Option 2)

    an ellipse

  • Option 3)

    a hyperbola

  • Option 4)

    a parabola.


Answers (1)

As we learnt in 

Condition for Tangency in Hyperbola -

C^{2}= a^{2}m^{2}-b^{2}

- wherein

For the Hyperbola

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


y = \alpha x + \beta is tangent to  \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1  

Condition for tangency for y = mn + C is C2= a2m2 - b2

i.e. \beta^{2}=a^{2}\alpha^{2}-b^{2}

Thus replace (\alpha , \beta ) by (x, y)

y2 = a2x2 - b2 which is hyperbola.  

Option 1)

a circle

this is incorrect option

Option 2)

an ellipse

this is incorrect option

Option 3)

a hyperbola

this is correct option

Option 4)

a parabola.

this is incorrect option

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