Question

Let $C$ be the circle of unit radius centred at the origin if two positive numbers $x_1$ and $x_2$ are such that the line passing though $(x_1-1)$ and $(x_2-1)$ is tangent to $C$ then

• A. $4x_1{x}_2=1$
• B. $x_1{x}_2=-1$
• C. $x_1+x_2=1$
• D. $x_1{x}_2=1$

Answer 1

The correct option is D $x_1{x}_2=1$

Eqn of tangent

$x\cos \theta +y\sin \theta =1$

put $(x_1,-1)$

$x_1\cos \theta -\sin \theta =1$

$x_1\cos \theta =1+\sin \theta$ .......... $\left(2\right)$

put $(x_2,1)$

$x_2\cos \theta =1-\sin \theta$

$x_1{x}_2{\cos }^2\theta =1-{\sin }^2\theta ={\cos }^2\theta$

$x_1{x}_2=1$