Question

In the figure, AC is tangent to the circle at O and CB is is the perpendicular from C to O. If the radius of the circle is r, then, select the statement that is true.

• A. $r^2=OB\times OA$
• B. $8r^2=OB\times OA$
• C. $7r^2=OB\times OA$
• D. $r^3=OB\times OA$

Answer 1

The correct option is A

$r^2=OB\times OA$

[Radius is perpendicular to the tangent at point of contact]

Join OC

OC = r

OC ⟂ AC

In △OCA and △OBC,

$\angle OCA=\angle OBC=90°$ [as each of them are 90°]

$\angle OCA=\angle OBC$ [Common angle]

△OCA ~ △OBC [AA similarity]

$\frac{OC}{OB}=\frac{OA}{OC}$

= $O{C}^2=OB\times OA$

$r^2=OB\times OA$