Maharashtra Board Class 10 Maths Chapter 3 Circle Set 3.2 Solutions

Question 1. Two circles having radii 3.5 cm and 4.8 cm touch each other internally. Find the distance between their centres.
Answer: Let the two circles having centres P and Q touch each other internally at point R. Here, QR = 3.5 cm, PR = 4.8 cm
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र दो वृत्तों को दर्शाता है जो आंतरिक रूप से एक-दूसरे को छूते हैं। बड़ा वृत्त जिसका केंद्र P है, छोटे वृत्त को छूता है जिसका केंद्र Q है, और दोनों वृत्त बिंदु R पर स्पर्श करते हैं। The two circles touch each other internally. \( \therefore \) By theorem of touching circles, P-Q-R PQ = PR – QR = 4.8 – 3.5 = 1.3 cm [The distance between the centres of circles touching internally is equal to the difference in their radii]
In simple words: When two circles touch internally, the distance between their centers is found by subtracting the radius of the smaller circle from the radius of the larger circle.

🎯 Exam Tip: Remember the theorem for internally touching circles: distance between centers = difference of radii. Draw a clear diagram to visualize the problem.

Question 2. Two circles of radii 5.5 cm and 4.2 cm touch each other externally. Find the distance between their centres.
Answer: Let the two circles having centres P and R touch each other externally at point Q. Here, PQ = 5.5 cm, QR = 4.2 cm
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र दो वृत्तों को दर्शाता है जो बाहरी रूप से एक-दूसरे को छूते हैं। एक वृत्त का केंद्र P है और दूसरे का केंद्र R है, और दोनों वृत्त बिंदु Q पर स्पर्श करते हैं। The two circles touch each other externally. \( \therefore \) By theorem of touching circles, P-Q-R PR = PQ + QR = 5.5 + 4.2 = 9.7 cm [The distance between the centres of the circles touching externally is equal to the sum of their radii]
In simple words: When two circles touch externally, the distance between their centers is found by adding their radii together.

🎯 Exam Tip: For externally touching circles, the distance between centers equals the sum of their radii. Always state the theorem used for justification.

Question 3. If radii of two circles are 4 cm and 2.8 cm. Draw figure of these circles touching each other
(i) externally
(ii) internally.
Answer:
(i) Circles touching externally:
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र दो वृत्त दर्शाता है जो बाहरी रूप से एक-दूसरे को छूते हैं। बड़े वृत्त का केंद्र P है और त्रिज्या 4 सेमी है, जबकि छोटे वृत्त का केंद्र R है और त्रिज्या 2.8 सेमी है। वे बिंदु Q पर स्पर्श करते हैं।
(ii) Circles touching internally:
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र दो वृत्त दर्शाता है जो आंतरिक रूप से एक-दूसरे को छूते हैं। बड़े वृत्त का केंद्र P है और त्रिज्या 4 सेमी है, जबकि छोटे वृत्त का केंद्र R है और त्रिज्या 2.8 सेमी है। वे बिंदु Q पर स्पर्श करते हैं।
In simple words: This question requires drawing two scenarios for circles with given radii: one where they touch externally (centers are far apart), and another where they touch internally (one circle inside the other).

🎯 Exam Tip: Practice drawing accurate diagrams for both external and internal touching circles. Label centers and radii clearly as specified in the problem.

Question 4. In the adjoining figure, the circles with centres P and Q touch each other at R A line passing through R meets the circles at A and B respectively. Prove that -
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र दो वृत्तों को दर्शाता है जिनके केंद्र P और Q हैं, जो बिंदु R पर एक-दूसरे को छूते हैं। एक सीधी रेखा R से गुजरती हुई पहले वृत्त को A पर और दूसरे वृत्त को B पर काटती है, जिससे एक AB रेखा बनती है।
(i) seg AP || seg BQ,
(ii) \( \triangle APR \sim \triangle RQB \), and
(iii) Find \( \angle RQB \) if \( \angle PAR = 35^\circ \).
Answer: The circles with centres P and Q touch each other at R. \( \therefore \) By theorem of touching circles, P-R-Q
(i) In \( \triangle PAR \), seg PA = seg PR [Radii of the same circle] \( \therefore \angle PRA = \angle PAR \) (i) [Isosceles triangle theorem] Similarly, in \( \triangle QBR \), seg QR = seg QB [Radii of the same circle] \( \therefore \angle RBQ = \angle QRB \) (ii) [Isosceles triangle theorem] But, \( \angle PRA = \angle QRB \) (iii) [Vertically opposite angles] \( \therefore \angle PAR = \angle RBQ \) (iv) [From (i) and (ii)] But, they are a pair of alternate angles formed by transversal AB on seg AP and seg BQ. \( \therefore \) seg AP || seg BQ [Alternate angles test]
(ii) In \( \triangle APR \) and \( \triangle RQB \), \( \angle PAR = \angle QRB \) [From (i) and (iii)] \( \angle APR = \angle RQB \) [Alternate angles] \( \therefore \triangle APR - \triangle RQB \) [AA test of similarity]
(iii) \( \angle PAR = 35^\circ \) [Given] \( \therefore \angle RBQ = \angle PAR = 35^\circ \) [From (iv)] In \( \triangle RQB \), \( \angle RQB + \angle RBQ + \angle QRB = 180^\circ \) [Sum of the measures of angles of a triangle is \( 180^\circ \)]
\( \implies \angle RQB + \angle RBQ + \angle RBQ = 180^\circ \) [From (ii)]
\( \implies \angle RQB + 2 \angle RBQ = 180^\circ \)
\( \implies \angle RQB + 2 \times 35^\circ = 180^\circ \)
\( \implies \angle RQB + 70^\circ = 180^\circ \)
\( \implies \angle RQB = 110^\circ \)
In simple words: This problem involves proving geometric properties using theorems of touching circles, isosceles triangles, vertically opposite angles, and the AA test for similarity. The final part uses the angle sum property of a triangle to find an unknown angle.

🎯 Exam Tip: Break down complex proofs into smaller steps. Clearly state the theorem or reason for each step. For similarity proofs, ensure corresponding angles are correctly identified.

Question 5. In the adjoining figure, the circles with centres A and B touch each other at E. Line l is a common tangent which touches the circles at C and D respectively. Find the length of seg CD if the radii of the circles are 4 cm, 6 cm.
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र दो वृत्तों को दर्शाता है जिनके केंद्र A और B हैं, जो एक-दूसरे को बिंदु E पर छूते हैं। एक उभयनिष्ठ स्पर्शरेखा 'l' पहले वृत्त को C पर और दूसरे वृत्त को D पर स्पर्श करती है। Construction: Draw seg AF \( \perp \) seg BD.
Answer:
(i) The circles with centres A and B touch each other at E. [Given] \( \therefore \) By theorem of touching circles, A-E-B
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र पिछले प्रश्न के समान है, लेकिन इसमें एक अतिरिक्त रचना दिखाई गई है: केंद्र A से रेखा BD पर एक लंब AF खींचा गया है, जो इस समस्या को हल करने में मदद करेगा। \( \therefore \angle ACD = \angle BDC = 90^\circ \) [Tangent theorem] \( \angle AFD = 90^\circ \) [Construction] \( \therefore \angle CAF = 90^\circ \) [Remaining angle of JAFDC] \( \therefore \) JAFDC is a rectangle. [Each angle is of measure \( 90^\circ \)] \( \therefore \) AC = DF = 4 cm [Opposite sides of a rectangle] Now, BD = BF + DF [B-F-C] \( \therefore \) 6 = BF + 4 BF = 2 cm Also, AB = AE + EB = 4 + 6 = 10 cm [The distance between the centres of circles touching externally is equal to the sum of their radii]
(ii) Now, in \( \triangle AFB \), \( \angle AFB = 90^\circ \) [Construction] \( \therefore AB^2 = AF^2 + BF^2 \) [Pythagoras theorem] \( \therefore 10^2 = AF^2 + 2^2 \) \( \therefore 100 = AF^2 + 4 \) \( \therefore AF^2 = 96 \) \( \therefore AF = \sqrt{96} \) [Taking square root of both sides] \( = \sqrt{16 \times 6} \) \( = 4 \sqrt{6} \) cm But, CD = AF [Opposite sides of a rectangle] \( \therefore CD = 4 \sqrt{6} \) cm
In simple words: To find the length of the common tangent segment CD, we use the property of externally touching circles and construct a perpendicular from one center to the radius of the other. This forms a rectangle and a right-angled triangle, allowing us to use the Pythagorean theorem to find the length of AF, which is equal to CD.

🎯 Exam Tip: For problems involving common tangents, constructing a perpendicular from one center to the radius of the other circle (or its extension) often creates a rectangle and a right triangle, making the Pythagorean theorem applicable. Clearly state your construction and the properties used.

Question 1. Take three collinear points X – Y – Z as shown in figure. Draw a circle with centre X and radius XY. Draw another circle with centre Z and radius YZ. Note that both the circles intersect each other at the single point Y. Draw a line l through point Y and perpendicular to seg XZ. What is line l (Textbook pg. no. 56)
Answer:
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र तीन संरेख बिंदुओं X, Y, Z को दर्शाता है। बिंदु X पर केंद्रित एक वृत्त जिसकी त्रिज्या XY है, और बिंदु Z पर केंद्रित एक वृत्त जिसकी त्रिज्या YZ है, दोनों एक-दूसरे को बिंदु Y पर छूते हैं। एक रेखा 'l' बिंदु Y से होकर गुजरती है और XZ के लंबवत है। Line l is a common tangent of the two circles.
In simple words: By constructing circles with centers X and Z and radii XY and YZ respectively, where X, Y, Z are collinear, we observe they touch at Y. The line passing through Y and perpendicular to XZ (the line connecting centers) is the common tangent to both circles.

🎯 Exam Tip: Understand that when two circles touch at a point and their centers are collinear, the line perpendicular to the line connecting their centers at the point of contact is their common tangent.

Question 2. Take points Y – X – Z as shown in the figure. Draw a circle with centre Z and radius ZY. Also draw a circle with centre X and radius XY. Note that both the circles intersect each other at the point Y. Draw a line l perpendicular to seg YZ through point Y. What is line l? (Textbook pg. no. 56)
Answer:
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र तीन संरेख बिंदुओं Y, X, Z को दर्शाता है। बिंदु Z पर केंद्रित एक वृत्त जिसकी त्रिज्या ZY है, और बिंदु X पर केंद्रित एक वृत्त जिसकी त्रिज्या XY है, दोनों एक-दूसरे को बिंदु Y पर आंतरिक रूप से छूते हैं। एक रेखा 'l' बिंदु Y से होकर गुजरती है और YZ के लंबवत है। Line l is a common tangent of the two circles. If two circles in the same plane intersect with a line in the plane in only one point, they are said to be touching circles and the line is their common tangent. The point common to the circles and the line is called their common point of contact.
In simple words: This activity demonstrates internally touching circles. When X is between Y and Z, and circles are drawn with centers Z (radius ZY) and X (radius XY), they touch at Y. The line perpendicular to the line connecting their centers at Y is their common tangent.

🎯 Exam Tip: This question reinforces the concept of a common tangent for internally touching circles. The line of centers (ZY) is crucial, and the tangent is perpendicular to it at the point of contact (Y).

1. Circles touching externally: For circles touching externally, the distance between their centres is equal to sum of their radii, i.e. AB = AC + BC
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र दो वृत्तों को दर्शाता है जो बाहरी रूप से एक-दूसरे को छूते हैं। उनके केंद्र A और B हैं, और वे एक उभयनिष्ठ बिंदु C पर स्पर्श करते हैं।
In simple words: When circles touch externally, their centers and the point of contact are collinear, and the distance between their centers is the sum of their individual radii.

🎯 Exam Tip: This is a fundamental definition. Remember, the centers and the point of contact always lie on the same straight line for touching circles.

2. Circles touching internally: For circles touching internally, the distance between their centres is equal to difference of their radii, i.e. AB = AC – BC
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र दो वृत्तों को दर्शाता है जो आंतरिक रूप से एक-दूसरे को छूते हैं। बड़े वृत्त का केंद्र C है और छोटे वृत्त का केंद्र B है। दोनों वृत्त एक उभयनिष्ठ बिंदु A पर स्पर्श करते हैं।
In simple words: When circles touch internally, their centers and the point of contact are collinear, and the distance between their centers is the absolute difference between their radii.

🎯 Exam Tip: Be careful with internal touching circles; the distance between centers is always the difference, and the centers are collinear with the point of contact.

Question 3. The circles shown in the given figure are called externally touching circles. Why?
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र दो वृत्तों को दर्शाता है जिनके केंद्र R और S हैं। वे एक सीधी रेखा 'l' द्वारा बिंदु T पर बाहरी रूप से स्पर्श करते हैं।
Answer: Circles with centres R and S lie in the same plane and intersect with a line l in the plane in one and only one point T [R – T – S]. Hence the given circles are externally touching circles.
In simple words: These are externally touching circles because they touch at only one point (T) and their centers (R, S) are on opposite sides of the common tangent line 'l' passing through T.

🎯 Exam Tip: The key identifiers for externally touching circles are a single point of contact and their centers lying on opposite sides of the tangent through that point.

Question 4. The circles shown in the given figure are called internally touching circles, why?
ℹ️ चित्र व्याख्या (Diagram Explanation): यह चित्र दो वृत्तों को दर्शाता है जिनके केंद्र N और M हैं। वे एक सीधी रेखा 'p' द्वारा बिंदु K पर आंतरिक रूप से स्पर्श करते हैं।
Answer: Circles with centres N and M lie in the same plane and intersect with a line p in the plane in one and only one point T [K – N – M]. Hence, the given circles are internally touching circles.
In simple words: These are internally touching circles because one circle is completely contained within the other, and they touch at only one common point (K) where their centers (N, M) and the point of contact are collinear.

🎯 Exam Tip: For internally touching circles, one center lies within the other circle, and they share exactly one point of contact, with a common tangent at that point.

Question 5. In the given figure, the radii of the circles with centres A and B are 3 cm and 4 cm respectively. Find
(i) d(A,B) in figure (a)
(ii) d(A,B) in figure (b)
ℹ️ चित्र व्याख्या (Diagram Explanation): (a) यह चित्र दो वृत्तों को दर्शाता है जिनके केंद्र A और B हैं, जो बिंदु C पर बाहरी रूप से स्पर्श करते हैं। (b) यह चित्र दो वृत्तों को दर्शाता है जिनके केंद्र A और B हैं, जो बिंदु C पर आंतरिक रूप से स्पर्श करते हैं।
Answer:
(i) Here, circle with centres A and B touch each other externally at point C. \( \therefore \) d(A, B) = d(A, C) + d(B,C) = 3 + 4 \( \therefore \) d(A,B) = 7 cm [The distance between the centres of circles touching externally is equal to the sum of their radii]
(ii) Here, circle with centres A and B touch each other internally at point C. \( \therefore \) d(A, B) = d(A, C) – d(B, C) = 4-3 \( \therefore \) d(A,B) = 1 cm [The distance between the centres of circles touching internally is equal to the difference in their radii]
In simple words: For externally touching circles (figure a), the distance between centers is the sum of their radii. For internally touching circles (figure b), the distance between centers is the difference of their radii.

🎯 Exam Tip: This question tests the direct application of theorems for distances between centers of touching circles. Ensure to identify whether the circles touch externally or internally before applying the formula.