Question

Write True or False:

The angle between two tangents to a circle may be $0^0$.

• A. True
• B. False
• C. Ambiguous
• D. Data insufficient

Answer 1

Answer: (A)

True

Given-

PA & PB are tangents to the circle with centre O at A & B.

To find out-

The statement, $\angle P=0^o$ is true or false.

Justification-

PA & PB are tangents to the circle at A & B.

$\therefore \angle PAO=\angle PBO={90}^o$

$\therefore \angle PAO+\angle PBO=2\times {90}^o={180}^o$ .......(i)

Now, in $\square PAOB$,

$\angle APB+\angle AOB+(\angle PAO+\angle PBO)={360}^o$ .....(angle sum property of quadrilaterals)

$⟹\angle APB+\angle AOB={180}^o$ .....(from i)

$\therefore$ Each of $\angle P$ & $\angle O$ is $<{180}^o$.....(ii)

So, if $\angle P=0^o$, then $\angle O={180}^o$ which contradicts (ii)

$\therefore \angle P$ can not be $=0^o$ unless P is at infinity.

$\angle P$ may be$=0^o$