Question

If $x\cos \alpha +y\sin \alpha =p$ is a tangent to the circle $x^2+y^2=2q(x\cos \alpha +y\sin \alpha )$ then the set of possible values of p is

• A. $\left\{q\right\}$
• B. $\{0,q\}$
• C. $\{0,2q\}$
• D. $\{q,2q\}$

Answer 1

The correct option is C $\{0,2q\}$

Centre and radius of the circle are, $(q\cos \alpha ,q\sin \alpha )$ and $r=\sqrt{q^2{\cos }^2\alpha +q^2{\sin }^2\alpha }=q$
Now for given line to be tangent to the circle.
$\left|\frac{q-p}{1}\right|=r=q\Rightarrow q-p=\pm q\Rightarrow p=0,2q$