CAT 2025 Lesson : Class Discussion: Geometry - Circles 9 to 16

Class Discussion - Circles

Question 9

A circle with radius 2 is placed against a right angle. Another smaller circle is also placed as shown in the adjoining figure. What is the radius of the smaller circle?
[CAT 2004]

		$3 - 2\sqrt2$
		$4 - 2\sqrt2$
		$7 - 4\sqrt2$
		$6 - 4\sqrt2$

Question 10

What is the distance in cm between two parallel chords of lengths 32 cm and 24 cm in a circle of radius 20 cm?
[CAT 2005]

		1 or 7
		2 or 14
		3 or 21
		4 or 28

Question 11

In a city, there is a circular park. There are four points of entry into the park, namely - P, Q, R and S. Three paths were constructed which connected the points PQ, RS, and PS. The length of the path PQ is 10 units, and the length of the path RS is 7 units. Later, the municipal corporation extended the paths PQ and RS past Q and R respectively, and they meet at a point T on the main road outside the park. The path from Q to T measures 8 units, and it was found that the angle PTS is 60 . Find the area (in square units) enclosed by the paths PT, TS, and PS.
[XAT 2011]

		$36\sqrt3$
		$54\sqrt3$
		$72\sqrt3$
		$90\sqrt3$
		None of the above

Question 12

Carpenter Rajesh has a circular piece of plywood of diameter 30 feet. He has cut out two disks of diameter 20 feet and 10 feet. What is the diameter of the largest disk that can be cut out from the remaining portion of the plywood piece?
[XAT 2012]

		$> 8.00$ ft and $≤ 8.20$ ft
		$> 8.21$ ft and $≤ 8.40$ ft
		$> 8.41$ ft and $≤ 8.60$ ft
		$> 8.61$ ft and $≤ 8.80$ ft
		$> 8.81$ ft and $≤ 9.00$ ft

Question 13

There are two circles

C_1

and

C_2

of radii 3 and 8 units respectively. The common internal tangent, T, touches the circles at points P1 and P2 respectively. The line joining the centres of the circles intersects T at X. The distance of X from the centre of the smaller circle is 5 units. What is the length of the line segment

P_{1}P_{2}

?
[XAT 2014]

		$< 13$
		$> 13$ and $< 14$
		$> 14$ and $< 15$
		$> 15$ and $< 16$
		$> 16$

Question 14

Two circles with radius 2R and

\sqrt2

R intersect each other at points A and B. The centers of both the circles are on the same side of AB. O is the center of the bigger circle and

\angle

AOB is

60 \degree

. Find the area of the common region between two circles.
[XAT 2018]

		$(\sqrt3 - \pi - 1)R^2$
		$(\sqrt3 - \pi )R^2$
		$(\dfrac{13 \pi}{6} + 1 - \sqrt3)R^2$
		$(\dfrac{13 \pi}{6} + 1 - \sqrt3)R^2$
		None of the above

Question 15

In the figure below, the circle has a chord AB of length 12 cm, which makes an angle of

60 \degree

at the center of the circle, O. ABCD, as shown in the diagram, is a rectangle. OQ is the perpendicular bisector of AB, intersecting the chord AB at P, the arc AB at M and CD at Q. OM = MQ. The area of the region enclosed by the line segments AQ and QB, and the arc BMA, is closest to (in

cm^2

):
[XAT 2020]

		35
		63
		69
		137
		215

Question 16

In a circle with center O and radius 1 cm, an arc AB makes an angle 60 degrees at O. Let R be the region bounded by the radii OA, OB and the arc AB. If C and D are two points on OA and OB, respectively, such that OC = OD and the area of triangle OCD is half that of R, then the length of OC, in cm, is
[CAT 2018 S1]

		$\left(\dfrac{\pi}{4\sqrt3}\right)^{\frac{1}{2}}$
		$\left(\dfrac{\pi}{4}\right)^{\frac{1}{2}}$
		$\left(\dfrac{\pi}{3\sqrt3}\right)^{\frac{1}{2}}$
		$\left(\dfrac{\pi}{6}\right)^{\frac{1}{2}}$

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