Circles Set-1

Q1. A chord of a circle is $16\ \text{cm}$ long and is at a distance $6\ \text{cm}$ from the centre. What is the radius of the circle?

Q2. From a point $P$ outside a circle a secant intersects the circle at $A$ and $B$ (with $P$–$A$–$B$ in that order) such that $PA=2\ \text{cm}$ and $PB=8\ \text{cm}$. What is the length of the tangent from $P$ to the circle?

Q3. Two parallel chords of a circle have lengths $10\ \text{cm}$ and $6\ \text{cm}$. The distance between the two chords is $4\ \text{cm}$. What is the radius of the circle?

Q4. Two circles of radii $5\ \text{cm}$ and $3\ \text{cm}$ have their centres $13\ \text{cm}$ apart. A common external tangent touches them at $T_1$ and $T_2$. What is the distance $T_1T_2$ between the points of contact?

Q5. A circle is tangent to both the $x$–axis and the $y$–axis and passes through the point $(1,2)$. Which of the following gives the possible radii of such a circle?

Q6. A circle has centre $O$ and radius $5\text{ cm}$. From an external point $P$ the distance $OP$ is $13\text{ cm}$. What is the length of the tangent from $P$ to the circle?

Q7. From a point $P$ outside a circle two secants $PAB$ and $PCD$ meet the circle at $A,B$ and $C,D$ respectively (with $A,C$ nearer to $P$). Given $PA=4\text{ cm}$, $PB=12\text{ cm}$ and $PC=6\text{ cm}$, find the length of the chord $CD$. (Use the secant power relation $PA\cdot PB=PC\cdot PD$.)

Q8. A circle of radius $13\text{ cm}$ has two parallel chords of lengths $10\text{ cm}$ and $24\text{ cm}$, both on the same side of the centre. What is the distance between these two chords?

Q9. Two circles of radii $5\text{ cm}$ and $3\text{ cm}$ have their centres $7\text{ cm}$ apart. The length of a direct common tangent segment between the two points of contact is:

Q10. Triangle $ABC$ has side lengths $AB=13\text{ cm}$, $BC=14\text{ cm}$ and $CA=15\text{ cm}$. Its circumradius $R$ equals:

Q11. A circle has radius $10\text{ cm}$. A chord is at a distance $6\text{ cm}$ from the centre. What is the length of the chord?

Q12. In a circle two chords $AB$ and $CD$ intersect at point $E$ inside the circle. If $AE=4x-1,\; EB=3x+5,\; CE=2x+3,\; ED=x+9$, find the length of chord $AB$.

Q13. From an external point $P$ two secants meet a circle at $A,B$ and $C,D$ respectively with $P$ nearer to $A$ and $C$. If $PA=4,\; PB=9,\; PC=x,\; PD=x+5$, find $PC$.

Q14. In a circle, chord $AB=12$. Tangents at $A$ and $B$ meet at $T$. If the distance $OT=13$, where $O$ is the centre of the circle, then the possible radius(es) of the circle are:

Q15. Two circles with centres $O_1(0,0)$ and $O_2(8,0)$ have radii $5$ and $3$ respectively. A common external tangent touches them at $X$ and $Y$. The length $XY$ of the segment between the points of contact is: