*Friday, May 10th, 2019*

In the earlier post on circles, we had discussed the properties and some sample CAT questions related to circles. In this post, we will see some additional CAT questions which have been asked in the previous years. Let us look at below examples.

** Example 1:** A one rupee coin is placed on a piece of paper. How many more coins of the same size may be placed such that each touches the central coin and the two adjacent coins?

a.) 7

b.) 4

c.) 5

d.) 6

** Solution:** It can be seen in the below figure that if we place 3 coins touching each other, their centers form an equilateral triangle. Hence, the angle made by the centers of the coins around the central coin is 60 degrees. Since the total angle to be covered is 360 degree, there has to be six coins surrounding the central coin.

** Example 2:** Three identical cones with base radius r are placed on their bases so that each in touching the other two. The radius of the circle drawn through their vertices is:

a.) Smaller than r

b.) Equal to r

c.) Larger than r

d.) Depends on the height of the cones

** Solution:** It can be seen from the below image that , if we place the 3 cones in such a way that they touch each other, it will be similar to placing three circles touching, with vertices of the cone corresponding to the centers of the circles. The centers of the circle form an equilateral triangle with each side being 2r. A circle that passes through the centers will be the circumcircle to such a triangle. The radius of the circumcircle of an equilateral triangle is (1/√3) times its side.

Hence, in our case it would be (2r/√3) and (2r/√3) > r, since √3 = 1.73 (approx)

Answer: Larger than r. Option c.

** Example 3:** Line AB is 6 meters in length and is tangent to the inner circle of the two concentric circles at point C. It is known that the radii of the two circles are integers. Find the radius of the outer circle.

a.) 5 meters

b.) 6 meters

c.) 7 meters

d.) 8 meters

** Solution:** Let x meters and y meters be the radius of the outer and inner circles respectively and O be their center.

In right angled triangle OCB,

CB^{2}= OB^{2} – OC^{2}

- 9 = x
^{2}– y^{2} - (x + y) (x – y)= 9 * 1

As x and y are integers, therefore x + y = 9 and x – y = 1.

Thus, x = 5

Hence radius of the outer circle is 5 meters.

** Example 4:** The figure shows a rectangle ABCD with a semicircle and a circle inscribed in it as shown. What is the ratio of the area of the circle to that of the semicircle?

** Solution:** Let the radius of the semicircle be R and radius of circle be r. Let P be the center of semicircle and Q be the center of the circle.

Now, Draw QS parallel to BC.

△PQS ~ △PBC

Therefore, PQ/PB = QS/BC

- (R + r)/ (√2*R) = (R-r)/R
- r = R (√2 – 1)
^{2}

Required ratio = (πr^{2}/ πR^{2})*2

Substituting r = R (√2 – 1)^{2} in above equation

Required ratio = 2 (√2 – 1)^{4 }: 1

** Example 5:** In the adjoining figure, points A, B, C and D lie on the circle. If AD = 24 and BC = 12, then what is the ratio of the area of the triangle CBE to that of the triangle ADE.

a.) 1 : 4

b.) 1 : 2

c.) 1 : 3

d.) Insufficient data

** Solution:** In △BEC and △AED

Angle CBE = Angle CDE (Since, angles in the same segment of a circle are equal)

Angle BEC = Angle AED (Vertical angles are equal)

Therefore, by AAA similarity

△CEB ~ △AED

We know that the ratio of areas of two similar triangles is equal to the ratio of squares of corresponding sides.

Therefore, (Area △CEB ) / (Area of △AED ) = (BC/AD)^{2 =} (12/24)^{2 }= ¼

** Example 6:** The sum of the areas of two circles which touch each other externally is 153π. If the sum of their radii is 15, find the ratio of the larger to the smaller radius.

a.) 4

b.) 2

c.) 3

d.) None of these

** Solution:** Let the radii of the 2 circles be r

and πr

⇒r

Solving we get, r

Ratio of the larger radius to the smaller one is 12 : 3 = 4 : 1 hence option (1) is the answer.

** Example 7:** Consider a circle with unit radius .There are seven adjacent sectors S₁, S₂ , S₃ , …..S₇ in the circle such that their total area is 1/8 of the area of the circle. Further the area of the ‘j’th sector is twice that of (j – 1) sector, for j = 2,….,7 what is the angle in radians, subtended by the arc of S₁, at the center of the circle.

a.) π/508

b.) π/2040

c.) π/1016

d.) π/1524

** Solution:** Let the area of sector S₁ be x units. Then the areas of corresponding sectors shall be 2x, 4x, 8x, 16x, 32x and 64x. Since every successive sector has an angle that is twice the previous one, the total area then shall be 127x units. This is 1/8 of the total area of the circle.

Hence the total area of the circle will be 127x * 8 units = 1016x units

Hence, angle of sector S₁ will be π/1016

** Example 8:** Find length of the common chord of two circles of radii 15 cm and 20 cm, distance between the centers being 25cm.

a.) 24 cm

b.) 25 cm

c.) 15 cm

d.) 20 cm

** Solution:** Let the length of the chord be x cm.

Therefore, from the diagram

½ (15 * 20) = ½ * 25 * x/2

⇒x = 24 cm

** Example 9:** In the figure given below ( not drawn to a scale), A, B, and C are 3 points on a circle with center 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 degrees and angle ACT = 50 degrees, then angle BOA is?

a.) 100 degrees

b.) 150 degrees

c.) 80 degrees

d.) Not possible to determine

__Solution:__

∠BAC = ∠ACT + ∠ATC = 50 ° + 30 ° = 80 °

And ∠ACT = ∠ABC (Angle in alternate segment)

So, ∠ABC = 50°

∠BCA = 180° – (∠ABC + ∠BAC)

= 180° – (50° + 80°)

= 50°

** Example 10:** In the adjoining figure, Chord ED is parallel to diameter AC . If angle CBE = 65° Determine angle CED

** Solution:** In triangle ABC, ∠B = 90 degrees (angle in a semicircle is a right angle)

Therefore ∠ABE = 90 – 65 = 25 degrees

Also ∠ABE= ∠ACE (Angle subtended by same arc AE)

Also, ∠ACE = ∠CED (AC || ED)

Therefore, ∠CED = 25 degree

** Conclusion: **Examples above are a good representation of the types of question asked in CAT examinations. If we are thorough with the concepts of circles (post 1) and these examples, solving questions related to circles should be pretty much doable. Cheers.

**Circles (Concepts, properties and CAT questions)**

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