# The locus of a point $\dpi{100} P(\alpha ,\beta )$ moving under the condition that the line $\dpi{100} y=\alpha x+\beta$ is a tangent to the hyperbola $\dpi{100} \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$   is Option 1) a circle Option 2) an ellipse Option 3) a hyperbola Option 4) a parabola.

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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