Bowls is a sport played on courts, which are flat and level grounds, limited by perimeter wooden boards. The objective of this sport is to throw bowls, which are balls made of synthetic material, in such a way as to place them as close as possible to the jack, which is a smaller ball made, preferably, of steel, previously thrown. Suppose that a player threw a bowl with radius 5 cm that ended up touching the jack with radius 2 cm, as illustrated in Figure 2.
Consider point $C$ as the center of the bowl, and point $O$ as the center of the jack. It is known that $A$ and $B$ are the points where the bowl and the jack, respectively, touch the ground of the court, and that the distance between $A$ and $B$ is equal to $d$. Under these conditions, what is the ratio between $d$ and the radius of the jack?
(A) 1
(B) $\frac{2\sqrt{10}}{5}$
(C) $\frac{\sqrt{10}}{2}$
(D) 2
(E) $\sqrt{10}$

129. Two circles with centers $O$ and $O'$ are externally tangent. A circle with diameter $\overline{OO'}$ is drawn with external common tangency to these two circles. What is the position of the two circles?

(1) Intersecting

(2) Tangent

(3) External

(4) Indeterminate

136- Circles $C$ and $C'$ are tangent at point $(0,1)$ and their common internal tangent lines with respect to circle $C$ are equidistant from point $(-3, 2)$. If circle $C'$ with radius $\sqrt{5}$ passes through $(-3,2)$, what is the center of circle $C'$?
(1) $(-1, 3)$ (2) $(-1, 2)$
(3) $(1, -2)$ (4) $(1, -1)$

136- A circle passing through point $(-9, -2)$ is tangent to both coordinate axes. What is the radius of the larger circle?
(1) $14$ (2) $15$ (3) $17$ (4) $19$

155. Suppose the length of the radical axis of two circles with radii $1 - 6a$ and $6 - 2a^2$ equals 6 units. If the two circles have exactly one common tangent, what is the average of the possible values of $a$?

• [(1)] $3$
• [(2)] $\dfrac{13}{3}$
• [(3)] $6$
• [(4)] $7$

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Let $C _ { 1 } , C _ { 2 }$ and $C _ { 3 }$ be three circles lying in the same quadrant, each touching both the axes. Suppose also that $C _ { 1 }$ touches $C _ { 2 }$ and $C _ { 2 }$ touches $C _ { 3 }$. If the area of the smallest circle is 1 unit, then area of the largest circle is
(a) $\{ ( \sqrt{2} + 1 ) / ( \sqrt{2} - 1 ) \} ^ { 4 }$
(b) $( 1 + \sqrt{2} ) ^ { 2 }$
(c) $( 2 + \sqrt{2} ) ^ { 2 }$
(d) $2 ^ { 4 }$

An isosceles triangle with base 6 cms. and base angles $30^\circ$ each is inscribed in a circle. A second circle touches the first circle and also touches the base of the triangle at its midpoint. If the second circle is situated outside the triangle, then its radius (in cms.) is
(A) $3\sqrt{3}/2$
(B) $\sqrt{3}/2$
(C) $\sqrt{3}$
(D) $4/\sqrt{3}$

Let $n \geq 3$ be an integer. Assume that inside a big circle, exactly $n$ small circles of radius $r$ can be drawn so that each small circle touches the big circle and also touches both its adjacent small circles. Then, the radius of the big circle is
(A) $r \operatorname{cosec} \frac{\pi}{n}$
(B) $r \left( 1 + \operatorname{cosec} \frac{2\pi}{n} \right)$
(C) $r \left( 1 + \operatorname{cosec} \frac{\pi}{2n} \right)$
(D) $r \left( 1 + \operatorname{cosec} \frac{\pi}{n} \right)$

Let $n \geq 3$ be an integer. Assume that inside a big circle, exactly $n$ small circles of radius $r$ can be drawn so that each small circle touches the big circle and also touches both its adjacent small circles. Then, the radius of the big circle is:
(a) $r \operatorname { cosec } \frac { \pi } { n }$
(b) $r \left( 1 + \operatorname { cosec } \frac { 2 \pi } { n } \right)$
(c) $r \left( 1 + \operatorname { cosec } \frac { \pi } { 2 n } \right)$
(d) $r \left( 1 + \operatorname { cosec } \frac { \pi } { n } \right)$.

An isosceles triangle with base 6 cms. and base angles $30 ^ { \circ }$ each is inscribed in a circle. A second circle, which is situated outside the triangle, touches the first circle and also touches the base of the triangle at its midpoint. Find its radius.

Let $n \geq 3$ be an integer. Assume that inside a big circle, exactly $n$ small circles of radius $r$ can be drawn so that each small circle touches the big circle and also touches both its adjacent small circles. Then, the radius of the big circle is:
(a) $r \operatorname { cosec } \frac { \pi } { n }$
(b) $r \left( 1 + \operatorname { cosec } \frac { 2 \pi } { n } \right)$
(c) $r \left( 1 + \operatorname { cosec } \frac { \pi } { 2 n } \right)$
(d) $r \left( 1 + \operatorname { cosec } \frac { \pi } { n } \right)$.

An isosceles triangle with base 6 cms. and base angles $30 ^ { \circ }$ each is inscribed in a circle. A second circle, which is situated outside the triangle, touches the first circle and also touches the base of the triangle at its midpoint. Find its radius.

An isosceles triangle with base 6 cms. and base angles $30^\circ$ each is inscribed in a circle. A second circle touches the first circle and also touches the base of the triangle at its midpoint. If the second circle is situated outside the triangle, then its radius (in cms.) is
(A) $3\sqrt{3}/2$
(B) $\sqrt{3}/2$
(C) $\sqrt{3}$
(D) $4/\sqrt{3}$

Let $n \geq 3$ be an integer. Assume that inside a big circle, exactly $n$ small circles of radius $r$ can be drawn so that each small circle touches the big circle and also touches both its adjacent small circles. Then, the radius of the big circle is
(A) $r \operatorname{cosec} \frac{\pi}{n}$
(B) $r \left( 1 + \operatorname{cosec} \frac{2\pi}{n} \right)$
(C) $r \left( 1 + \operatorname{cosec} \frac{\pi}{2n} \right)$
(D) $r \left( 1 + \operatorname{cosec} \frac{\pi}{n} \right)$

An isosceles triangle with base 6 cms. and base angles $30 ^ { \circ }$ each is inscribed in a circle. A second circle touches the first circle and also touches the base of the triangle at its midpoint. If the second circle is situated outside the triangle, then its radius (in cms.) is
(A) $3 \sqrt { 3 } / 2$
(B) $\sqrt { 3 } / 2$
(C) $\sqrt { 3 }$
(D) $4 / \sqrt { 3 }$

Let $n \geq 3$ be an integer. Assume that inside a big circle, exactly $n$ small circles of radius $r$ can be drawn so that each small circle touches the big circle and also touches both its adjacent small circles. Then, the radius of the big circle is
(A) $r \operatorname { cosec } \frac { \pi } { n }$
(B) $r \left( 1 + \operatorname { cosec } \frac { 2 \pi } { n } \right)$
(C) $r \left( 1 + \operatorname { cosec } \frac { \pi } { 2 n } \right)$
(D) $r \left( 1 + \operatorname { cosec } \frac { \pi } { n } \right)$

The centres of two circles $C_{1}$ and $C_{2}$ each of unit radius are at a distance of 6 units from each other. Let $P$ be the mid point of the line segment joining the centres of $C_{1}$ and $C_{2}$ and $C$ be a circle touching circles $C_{1}$ and $C_{2}$ externally. If a common tangent to $C_{1}$ and $C$ passing through $P$ is also a common tangent to $C_{2}$ and $C$, then the radius of the circle $C$ is

The circle $C_1: x^2 + y^2 = 3$, with centre at $O$, intersects the parabola $x^2 = 2y$ at the point $P$ in the first quadrant. Let the tangent to the circle $C_1$ at $P$ touches other two circles $C_2$ and $C_3$ at $R_2$ and $R_3$, respectively. Suppose $C_2$ and $C_3$ have equal radii $2\sqrt{3}$ and centres $Q_2$ and $Q_3$, respectively. If $Q_2$ and $Q_3$ lie on the $y$-axis, then
(A) $Q_2 Q_3 = 12$
(B) $R_2 R_3 = 4\sqrt{6}$
(C) area of the triangle $OR_2R_3$ is $6\sqrt{2}$
(D) area of the triangle $PQ_2Q_3$ is $4\sqrt{2}$

Let $A B C$ be the triangle with $A B = 1 , A C = 3$ and $\angle B A C = \frac { \pi } { 2 }$. If a circle of radius $r > 0$ touches the sides $A B , A C$ and also touches internally the circumcircle of the triangle $A B C$, then the value of $r$ is $\_\_\_\_$.

Let $G$ be a circle of radius $R > 0$. Let $G _ { 1 } , G _ { 2 } , \ldots , G _ { n }$ be $n$ circles of equal radius $r > 0$. Suppose each of the $n$ circles $G _ { 1 } , G _ { 2 } , \ldots , G _ { n }$ touches the circle $G$ externally. Also, for $i = 1,2 , \ldots , n - 1$, the circle $G _ { i }$ touches $G _ { i + 1 }$ externally, and $G _ { n }$ touches $G _ { 1 }$ externally. Then, which of the following statements is/are TRUE ?
(A) If $n = 4$, then $( \sqrt { 2 } - 1 ) r < R$
(B) If $n = 5$, then $r < R$
(C) If $n = 8$, then $( \sqrt { 2 } - 1 ) r < R$
(D) If $n = 12$, then $\sqrt { 2 } ( \sqrt { 3 } + 1 ) r > R$

Let the straight line $y = 2 x$ touch a circle with center $( 0 , \alpha ) , \alpha > 0$, and radius $r$ at a point $A _ { 1 }$. Let $B _ { 1 }$ be the point on the circle such that the line segment $A _ { 1 } B _ { 1 }$ is a diameter of the circle. Let $\alpha + r = 5 + \sqrt { 5 }$.
Match each entry in List-I to the correct entry in List-II.
List-I
(P) $\alpha$ equals
(Q) $r$ equals
(R) $A _ { 1 }$ equals
(S) $B _ { 1 }$ equals
List-II
(1) $( - 2,4 )$
(2) $\sqrt { 5 }$
(3) $( - 2,6 )$
(4) 5
(5) $( 2,4 )$
The correct option is
(A) $( \mathrm { P } ) \rightarrow ( 4 )$, $( \mathrm { Q } ) \rightarrow ( 2 )$, $( \mathrm { R } ) \rightarrow ( 1 )$, $( \mathrm { S } ) \rightarrow ( 3 )$
(B) $( \mathrm { P } ) \rightarrow ( 2 )$, $( \mathrm { Q } ) \rightarrow ( 4 )$, $( \mathrm { R } ) \rightarrow ( 1 )$, $( \mathrm { S } ) \rightarrow ( 3 )$
(C) $( \mathrm { P } ) \rightarrow ( 4 )$, $( \mathrm { Q } ) \rightarrow ( 2 )$, $( \mathrm { R } ) \rightarrow ( 5 )$, $( \mathrm { S } ) \rightarrow ( 3 )$
(D) $( \mathrm { P } ) \rightarrow ( 2 )$, $( \mathrm { Q } ) \rightarrow ( 4 )$, $( \mathrm { R } ) \rightarrow ( 3 )$, $( \mathrm { S } ) \rightarrow ( 5 )$

Consider a family of circles which are passing through the point $( - 1,1 )$ and are tangent to $x -$ axis. If $( h , k )$ are the co-ordinates of the centre of the circles, then the set of values of $k$ is given by the interval
(1) $0 < \mathrm { k } < 1 / 2$
(2) $k \geq 1 / 2$
(3) $- 1 / 2 \leq k \leq 1 / 2$
(4) $k \leq 1 / 2$

The two circles $x^{2}+y^{2}=ax$ and $x^{2}+y^{2}=c^{2}$ $(c>0)$ touch each other if
(1) $|a|=c$
(2) $a=2c$
(3) $|a|=2c$
(4) $2|a|=c$

If the circle $x ^ { 2 } + y ^ { 2 } - 6 x - 8 y + \left( 25 - a ^ { 2 } \right) = 0$ touches the axis of $x$, then $a$ equals.
(1) 0
(2) $\pm 4$
(3) $\pm 2$
(4) $\pm 3$

Let $C$ be the circle with center at $( 1,1 )$ and radius $= 1$. If $T$ is the circle centered at $( 0 , y )$, passing through the origin and touching the circle $C$ externally, then the radius of $T$ is equal to
(1) $\frac { 1 } { 2 }$
(2) $\frac { 1 } { 4 }$
(3) $\frac { \sqrt { 3 } } { \sqrt { 2 } }$
(4) $\frac { \sqrt { 3 } } { 2 }$