Question

If the line $ycos\alpha =xsin\alpha +acos\alpha$ be a tangent to the circle $x^2+y^2=a^2$,then

• A. $si{n}^2\alpha =1$
• B. $co{s}^2\alpha =1$
• C. $si{n}^2\alpha =a^2$
• D. $co{s}^2\alpha =a^2$

Answer 1

The correct option is B $co{s}^2\alpha =1$

$x\sin \alpha -y\cos \alpha +a\cos \alpha =0x^2+y^2=a^2$

Distance from center should be equal to radius, center$=(0,0)$, radius=a

$\left|\frac{0\left(\sin \alpha \right)-\left(0\right)\cos \alpha +a\cos \alpha }{\sqrt{{\sin }^2\alpha +{\cos }^2\alpha }}\right|=a$

$\left|a\cos \alpha \right|=a$

${\cos }^2\alpha =1$