Question

A standard hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is drawn along with its auxiliary circle. A point P $(asec\theta ,btan\theta )$ is taken. A perpendicular is dropped from P to the x axis which meets at the axis at R. A tangent is drawn from R to auxiliary circle. Which angle is equal to θ

• A. $\angle TOR$
• B. $\angle TRO$
• C. $\angle PRT$
• D. $\angle TO{V}_1$

Answer 1

The correct option is A

$\angle TOR$

The given situation pertains to the graphical interpretation of the parametric form of a hyperbola .

Let T$(x_1,y_1)$ be the point at which tangent touches the circle.

The tangent can be given by the equation,

$x{x}_1+y{y}_1=a^2$

This passes through R$(asec\theta ,0)$

$\therefore asec\theta .x_1+0=a^2$

$x_1=acos\theta$

$(x_1,y_1)$ is on the circle

i.e., $x_1^2+y_1^2=a^2$

$\therefore y_1=asin\theta$

$\therefore Q=(acos\theta ,asin\theta )$ where a is the radius of circle.This is true when $\angle TOR=\theta$