Class Discussion - Circles

Question 1

A certain city has a circular wall around it, and this wall has four gates pointing north, south, east and west. A house stands outside the city, three km north of the north gate, and it can just be seen from a point nine km east of the south gate. What is the diameter of the wall that surrounds the city?
[CAT 2001]

		6 km
		9 km
		12 km
		None of these

Question 2

Consider three circular parks of equal size with centres at $A_1$, $A_2$ and $A_3$ respectively. The parks touch each other at the edge as shown in the figure (not drawn to scale). There are three paths formed by the triangles $A_1$ $A_2$ $A_3$, $B_1$ $B_2$ $B_3$ and $C_1$ $C_2$ $C_3$, as shown. Three sprinters A, B and C begin running from points $A_1$, $B_1$ and $C_1$ respectively. Each sprinter traverses her respective triangular path clockwise and returns to her starting point.
[CAT 2003 Retest]

Let the radius of each circular park be $r$, and the distances to be traversed by the sprinters A, B and C be $a$, $b$ and $c$, respectively. Which of the following is true?

		$b - a = c - b = 3 \sqrt3 r$
		$b - a = c - b = \sqrt3 r$
		$b = \dfrac{a + c}{2} = 2(1 + \sqrt3) r$
		$c = 2b - a = (2 + \sqrt3)r$

Question 3

Consider three circular parks of equal size with centres at $A_1$, $A_2$ and $A_3$ respectively. The parks touch each other at the edge as shown in the figure (not drawn to scale). There are three paths formed by the triangles $A_1$ $A_2$ $A_3$, $B_1$ $B_2$ $B_3$ and $C_1$ $C_2$ $C_3$, as shown. Three sprinters A, B and C begin running from points $A_1$, $B_1$ and $C_1$ respectively. Each sprinter traverses her respective triangular path clockwise and returns to her starting point.
[CAT 2003 Retest]

Sprinter A traverses distances $A_{1}A_{2}$, $A_{2}A_{3}$ and $A_{3}A_{1}$ at average speeds of 20, 30 and 15 respectively. B traverses her entire path at a uniform speed of $(10 \sqrt3 + 20)$ . C traverses distances $C_{1}C_{2}$, $C_{2}C_{3}$ and $C_{3}C_{1}$ at average speeds of $\dfrac{40}{3} (\sqrt3 + 1)$, $\dfrac{40}{3} (\sqrt3 + 1)$, and 120 respectively. All speeds are in the same unit. Where would B and C be respectively when A finishes her sprint?

		$B_1, C_1$
		$B_3, C_3$
		$B_1, C_3$
		$B_1$, Somewhere between $C_3$ and $C_1$

Question 4

Consider three circular parks of equal size with centres at $A_1$, $A_2$ and $A_3$ respectively. The parks touch each other at the edge as shown in the figure (not drawn to scale). There are three paths formed by the triangles $A_1$ $A_2$ $A_3$, $B_1$ $B_2$ $B_3$ and $C_1$ $C_2$ $C_3$, as shown. Three sprinters A, B and C begin running from points $A_1$, $B_1$ and $C_1$ respectively. Each sprinter traverses her respective triangular path clockwise and returns to her starting point.
[CAT 2003 Retest]

Sprinters A, B and C traverse their respective paths at uniform speeds $u$, $v$ and $w$ respectively. It is known that $u^2$ : $v^2$ : $w^2$ is equal to Area A : Area B : Area C, where Area A, Area B and Area C are the areas of triangles $A_{1}A_{2}A_{3}$, $B_{1}B_{2}B_{3}$, and $C_{1}C_{2}C_{3}$ respectively. Where would A and C be when B reaches point $B_3$?

		$A_{2}, C_{3}$
		$A_{3}, C_{3}$
		$A_{3}, C_{2}$
		Somewhere between $A_{2}$ and $A_{3}$, Somewhere between $C_{3}$ and $C_{1}$

Question 5

The length of the circumference of a circle equals the perimeter of a triangle of equal sides, and also the perimeter of a square. The areas covered by the circle, triangle, and square are $c$, $t$, and $s$, respectively. Then,
[CAT 2003 Retest]

		$s > t > c$
		$c > t > s$
		$c > s > t$
		$s > c > t$

Question 6

In the figure below (not drawn to scale), rectangle ABCD is inscribed in the circle with centre at O. The length of side AB is greater than that of side BC. The ratio of the area of the circle to the area of the rectangle ABCD is $\pi : \sqrt3$. The line segment DE intersects AB at E such that $\angle$ODC = $\angle$ADE. What is the ratio AE : AD?
[CAT 2003 Retest]

		$1 : \sqrt3$
		$1 : \sqrt2$
		$1 : 2\sqrt3$
		1 : 2

Question 7

In the figure given below (not drawn to scale), A, B and C are three points on a circle with centre O. The chord BA is extended to a point T such that CT becomes a tangent to the circle at point C. If $\angle$ATC = $30 \degree$ and $\angle$ACT = $50 \degree$ , then the angle $\angle$BOA is:
[CAT 2003 Retest]

		$100 \degree$
		$150 \degree$
		$80 \degree$
		Cannot be determined

Question 8

Let C be a circle with centre $P_0$ and AB be a diameter of C. Suppose $P_1$ is the mid-point of the line segment $P_{0}B$, $P_2$ is the mid-point of the line segment $P_{1}B$ and so on. Let $C_1$, $C_2$, $C_3$, ... be circles with diameters $P_{0}P_{1}$, $P_{1}P_{2}$, $P_{2}P_{3}$ ... respectively. Suppose the circles $C_1$, $C_2$, $C_3$, ... are all shaded. The ratio of the area of the unshaded portion of C to that of the original circle C is:
[CAT 2004]

		8 : 9
		9 : 10
		10 : 11
		11 : 12