Circles: CBSE Class 10 Mathematics practice questions

B. $1$

A line that meets the circle in two points is a secant. As the second point slides towards $P$ the two points coincide, so a tangent is the special case of a secant with exactly one common point (the point of contact). A line with no common point is a non-intersecting line.

C. $3$

$A$ is inside ($3<5$): no tangent. $B$ is on the circle ($5=5$): one tangent. $C$ is outside ($8>5$): two tangents. Total $0+1+2=3$. The traps: treating every point as outside ($6$), thinking an outside point gives one tangent ($2$), or thinking an inside point gives one ($4$).

B. $6$ cm

Tangent lengths: $AC=AP=4$ and $BC=BR=9$. $OA$ and $OB$ bisect the angles at $A$ and $B$, whose sum is $180^\circ$, so $\angle AOB=90^\circ$ and $OC\perp AB$ is the altitude to the hypotenuse. Hence $OC^2=AC\times BC=36$ and $r=6$. Traps: the average $\frac{4+9}{2}=6.5$, the sum $13$, or $\sqrt{4^2+9^2}=\sqrt{97}$.