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Two tangents PQ and PR are drawn from an external point to a circle with centre O. Prove that QORP is a cyclic quadrilateral.

Radius ⊥ Tangent = OR ⊥ PR Hence ∠ORP = 90° Similarly OQ ⊥PQ ∠OQP = 90° In... View Article

Out of the two concentric circles, the radius of the outer circle is 5 cm and the chord AC of length 8 cm is a tangent to the inner circle. Find the radius of the inner circle.

Solution Chord AB = 8 cm OC is perpendicular to the chord AB =>OC⊥AB AC = CB = 4 cm From right triangle OCA OC2 +... View Article

If angle between two tangents drawn from a point P to a circle of radius a and centre O is 90°, then OP = a√2. Write ‘True’ or ‘False’ and justify your answer.

The given statement is True Solution Tangent is always perpendicular to the radius at the point of contact. Hence, ∠RPT = 90 We... View Article

The angle between two tangents to a circle may be 0°. Write ‘True’ or ‘False’ and justify your answer.

The given statement is True Solution The angle between two tangents to a circle may be 0°only when both tangent lines coincide or... View Article

The length of tangent from an external point P on a circle with centre O is always less than OP. Write ‘True’ or ‘False’ and justify your answer.

The given statement is True Solution Let us consider the figure of a circle with centre O. Let PT be a tangent drawn from... View Article

The length of tangent from an external point on a circle is always greater than the radius of the circle. Write ‘True’ or ‘False’ and justify your answer.

The given statement is False Solution Length of a tangent from an external point P on a circle may or may not be greater than the radius... View Article

If a chord AB subtends an angle of 60° at the centre of a circle, then angle between the tangents at A and B is also 60°. Write ‘True’ or ‘False’ and justify your answer.

The given statement is False Solution Let u consider the figure given below. In which we have a circle with centre O and AB a chord with... View Article

At one end A of a diameter AB of a circle of radius 5 cm, tangent XAY is drawn to the circle. The length of the chord CD parallel to XY and at a distance 8 cm from A is (A) 4 cm (B) 5 cm (C) 6 cm (D) 8 cm

The correct answer is (D) 8 cm Solution According to the given details Radius of circle, AO=OC = 5cm AM=8CM AM=OM+AO OM =AM-AO... View Article

From a point P which is at a distance of 13 cm from the centre O of a circle of radius 5 cm, the pair of tangents PQ and PR to the circle are drawn. Then the area of the quadrilateral PQOR is (A) 60 cm2 (B) 65 cm2 (C) 30 cm2 (D) 32.5 cm2

The correct answer is (A) 60 cm2 Solution Construction: Draw a circle of radius 5 cm with center O. Let P be a point at a distance of 13 cm... View Article

In Figure AB is a chord of the circle and AOC is its diameter such that ACB = 50°. If AT is the tangent to the circle at the point A, then BAT is equal to (A) 65° (B) 60° (C) 50° (D) 40°

The correct answer is (C) 50° Solution According to the given details A circle with centre O, diameter AC and ∠ACB = 50°... View Article

In Figure, if AOB = 125°, then COD is equal to (A) 62.5° (B) 45° (C) 35° (D) 55°

The correct answer is (D) 55° Solution ABCD is a quadrilateral circumscribing the circle We know that the opposite sides of a... View Article

If radii of two concentric circles are 4 cm and 5 cm, then the length of each chord of one circle which is tangent to the other circle is (A) 3 cm (B) 6 cm (C) 9 cm (D) 1 cm

The correct answer is (B) 6 cm Solution According to the given data we get to know that OA = 4cm OB = 5cm OA ⊥ BC... View Article

A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height h. At a point on the plane, the angles of elevation of the bottom and the top of the flag staff are α and β, respectively. Prove that the height of the tower is [h tan α/(tan β – tan α)].

Given Avertical flagstaff of height h is surmounted on a vertical tower of height H(say), such that FP = h and FO = H. The angle of... View Article

The shadow of a tower standing on a level plane is found to be 50 m longer when Sun’s elevation is 30° than when it is 60°. Find the height of the tower.

Let SQ = h be the tower. ∠SPQ = 30° and ∠SRQ = 60° According to the question, the length of the shadow is 50 m long hen... View Article

The angle of elevation of the top of a tower from two points distant s and t from its foot are complementary. Prove that the height of the tower is √st .

Let BC = s; PC = t Let height of the tower be AB = h. ∠ABC = θ and ∠APC = 90° – θ... View Article

Given that sinθ + 2cosθ = 1, then prove that 2sinθ – cosθ = 2.

Given sin θ +2 cos θ = 1 To Prove 2sinθ – cosθ = 2 Proof sinθ + 2cosθ = 1 On squaring... View Article

If 1 + sin2θ = 3sinθ cosθ , then prove that tanθ = 1 or 1/2.

Given 1+sin2 θ = 3 sin θ cos θ To Prove tanθ = 1 or 1/2 Proof 1+sin2 θ = 3 sin... View Article

The angle of elevation of the top of a tower from certain point is 30°. If the observer moves 20 metres towards the tower, the angle of elevation of the top increases by 15°. Find the height of the tower.

Let PR = h meter, be the height of the tower. The observer is standing at point Q such that, the distance between the observer and tower is QR =... View Article

Prove that √(sec2 θ + cosec2 θ) = tan θ + cot θ

To Prove √(sec2 θ + cosec2 θ) = tan θ + cot θ Proof Considering LHS √(sec2 θ +... View Article

If cosecθ + cotθ = p, then prove that cosθ = (p² – 1)/ (p² + 1).

Given cosecθ + cotθ = p To Prove cosθ = (p2 – 1)/ (p2 + 1) Proof cosec θ + cot θ = p (Given)... View Article

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