Class 10 Mathematics - Chapter Circles NCERT Solutions | If tangents PA and PB from a point P to

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

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.

How many tangents can a circle have?

Prove that the parallelogram circumscribing a circle is a rhombus.

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

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°

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                   .

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

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.

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.

Use Euclid’s division algorithm to find the HCF of :
   (i) 135 and 225      (ii) 196 and 38220      (iii) 867 and 255

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.

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).

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.

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                  . Such an event is called                       .
(iii) The probability of an event that is certain to happen is                  . Such an event is called                  .
(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                   and less than or equal to                  .

Check whether the following are quadratic equations :
(i) (x + 1)² = 2(x – 3) (ii) x² – 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) x²+ 3x + 1 = (x – 2)² (vii) (x + 2)³ = 2x (x2 – 1) (viii) x³ – 4x² – x + 1 = (x – 2)³

Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.

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.

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.

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.

Find the roots of the following quadratic equations, if they exist, by the method of
completing the square:
(i) 2x² – 7x + 3 = 0 (ii) 2x² + x – 4 = 0  (iv) 2x² + x + 4 = 0

The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.

A game consists of tossing a one rupee coin 3 times and noting its outcome each time. Hanif wins if all the tosses give the same result i.e., three heads or three tails, and loses otherwise. Calculate the probability that Hanif will lose the game.

Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following :
(i) p(x) = x³ – 3x² + 5x – 3, g(x) = x² – 2    (ii) p(x) = x⁴ – 3x² + 4x + 5, g(x) = x² + 1 – x      (iii) p(x) = x⁴ – 5x + 6, g(x) = 2 – x²

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(i) x² – 2x – 8 (ii) 4s² – 4s + 1 (iii) 6x² – 3 – 7x (iv) 4u² + 8u (v) t² – 15 (vi) 3x² – x – 4

A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is 60°. After some time, the angle of elevation reduces to 30° (see Fig. 9.13). Find the distance travelled by the balloon during the interval.

Find two consecutive positive integers, sum of whose squares is 365.

As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.

Is it possible to design a rectangular park of perimeter 80 m and area 400 m²? If so, find its length and breadth.