NCERT Solutions for Class 10 Maths Chapter 10 Circles वृत
Important Questions for Class 10 Maths Chapter 10 Circles
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Exercise 10.1
Question 1:
How many tangents can a circle have?
Question 2:
Fill in the blanks :
(i) A tangent to a circle intersects it in _______ point (s).
(ii) A line intersecting a circle in two points is called a _______ .
(iii) A circle can have _______ parallel tangents at the most.
(iv) The common point of a tangent to a circle and the circle is called _______ .
Question 3:
A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is :
(A) 12 cm \(\quad\) (B) 13 cm \(\quad\) (C) 8.5 cm \(\quad\) (D) \(\sqrt{119}\) cm
Question 4:
Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.
NCERT Solutions for Class 10 Maths Chapter 10 Circles
Exercise 10.2
Question 1:
From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is
(A) 7 cm (B) 12 cm (C) 15 cm (D) 24.5 cm
Question 2:
In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that \(\angle\) POQ = 110°, then \(\angle\) PTQ is equal to:
(A) 60° (B) 70° (C) 80° (D) 90°
Question 3:
If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then \(\angle\) POA is equal to
(A) 50° (B) 60° (C) 70° (D) 80°
Question 4:
Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
Question 5:
Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
Question 6:
The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.
Question 7:
Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
दो संकेंद्रीय वृतों की त्रिज्याएँ 5 cm और 3 cm हैं। बड़े वृत की जीवा, जो छोटे वृत को स्पर्श करती है, की लंबाई ज्ञात कीजिए।
Question 8:
A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove that AB + CD = AD + BC
Question 9:
In Fig. 10.13, XY and X’Y’ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X’Y’ at B. Prove that \(\angle\) AOB = 90°.
Question 10:
Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.
Question 11:
Prove that the parallelogram circumscribing a circle is a rhombus.
सिद्ध कीजिए कि किसी वृत्त के परिगत समांतर चतुर्भुज समचतुर्भुज होता है।
Solution
∴ AB || CD and BC || AD.
∴ AB = CD and BC = AD
The lengths of tangents drawn from an external point to a circle are equal.
Therefore, ⇒ BP = BQ ……….. (1)
⇒ CR = CQ ……….. (2)
⇒ DR = DS ……….. (3)
⇒ AP = AS ……….. (4)
Adding (1) + (2) + (3) + (4), we get :
⇒ BP + CR + DR + AP = BQ + CQ + DS + AS
⇒ (BP + AP) + (CR + DR) = (BQ + CQ) + (DS + AS)
⇒ AB + CD = BC + AD
Substitute CD = AB and AD = BC since ABCD is a parallelogram, then
⇒ AB + AB = BC + BC
⇒ 2AB = 2BC
⇒ AB = BC
∴ AB = BC = CD = DA
This implies that all the four sides are equal.
Hence, proved that the parallelogram circumscribing a circle is a rhombus.
Question 12:
A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively (see Fig. 10.14). Find the sides AB and AC.
Question 13:
Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
NCERT Solutions for Class 10 Maths Chapter 10 Circles
