Welcome to the NCERT Solutions for Class 10th Mathematics - Chapter Circles. This page offers a step-by-step solution to the specific question from Exercise 2, Question 11: **prove that the parallelogram circumscribing a circ...**.

Question 11

Prove that the parallelogram circumscribing a circle is a rhombus.

Answer

- Q:-
Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

- Q:-
How many tangents can a circle have?

- Q:-
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.

- Q:-
Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

- Q:-
Fill in the blanks :

(i) A tangent to a circle intersects it in

(ii) A line intersecting a circle in two points is called a

(iii) A circle can have

(iv) The common point of a tangent to a circle and the circle is called - Q:-
In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that ∠ POQ = 110°, then ∠ PTQ is equal to

(A) 60° (B) 70°

(C) 80° (D) 90° - Q:-
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 ∠ POA is equal to

(A) 50° (B) 60°

(C) 70° (D) 80° - Q:-
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 - Q:-
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.

- Q:-
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.

- Q:-
Use Euclid’s division algorithm to find the HCF of :

(i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255 - Q:-
The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

- Q:-
A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30° (see Fig. 9.11).

- Q:-
Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be.” (Isn’t this interesting?) Represent this situation algebraically and graphically.

- Q:-
Complete the following statements:

(i) Probability of an event E + Probability of the event ‘not E’ =

(ii) The probability of an event that cannot happen is

(iii) The probability of an event that is certain to happen is

(iv) The sum of the probabilities of all the elementary events of an experiment is(v) The probability of an event is greater than or equal to

- Q:-
Check whether the following are quadratic equations :

(i) (x + 1)^{2}= 2(x – 3) (ii) x^{2}– 2x = (–2) (3 – x) (iii) (x – 2)(x + 1) = (x – 1)(x + 3) (iv) (x – 3)(2x +1) = x(x + 5)(v) (2x – 1)(x – 3) = (x + 5)(x – 1) (vi) x

^{2}+ 3x + 1 = (x – 2)^{2}(vii) (x + 2)^{3}= 2x (x2 – 1) (viii) x^{3}– 4x^{2}– x + 1 = (x – 2)^{3} - Q:-
Show that any positive odd integer is of the form 6

*q*+ 1, or 6*q*+ 3, or 6*q*+ 5, where*q*is some integer. - Q:-
A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.

- Q:-
The coach of a cricket team buys 3 bats and 6 balls for ` 3900. Later, she buys another bat and 3 more balls of the same kind for ` 1300. Represent this situation algebraically and geometrically.

- Q:-
Which of the following experiments have equally likely outcomes? Explain.

(i) A driver attempts to start a car. The car starts or does not start.

(ii) A player attempts to shoot a basketball. She/he shoots or misses the shot.

(iii) A trial is made to answer a true-false question. The answer is right or wrong.

(iv) A baby is born. It is a boy or a girl.

- Q:-
Find the roots of the following equations:

- Q:-
Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:

(i) 2x

^{2 }– 3x + 5 = 0 (iii) 2x^{2}– 6x + 3 = 0 - Q:-
A game of chance consists of spinning an arrow which comes to rest pointing at one of the numbers 1, 2, 3, 4, 5, 6, 7, 8 (see Fig. 15.5 ), and these are equally likely outcomes. What is the probability that it will point at

(i) 8 ?

(ii) an odd number?

(iii) a number greater than 2?

(iv) a number less than 9? - Q:-
Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be.” (Isn’t this interesting?) Represent this situation algebraically and graphically.

- Q:-
The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

- Q:-
Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m

^{2}? If so, find its length and breadth. - Q:-
Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, –7, –14 respectively.

- Q:-
One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting

(i) a king of red colour (ii) a face card (iii) a red face card

(iv) the jack of hearts (v) a spade (vi) the queen of diamonds - Q:-
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(i) x

^{2}– 2x – 8 (ii) 4s^{2}– 4s + 1 (iii) 6x^{2}– 3 – 7x (iv) 4u^{2}+ 8u (v) t^{2 }– 15 (vi) 3x^{2 }– x – 4 - Q:-
Prove that 3 + 2√5 is irrational.

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