Intersection between Tangent and Secant

Q. The common tangents to the circle $x^2+y^2=2$ and the parabola $y^2=8x$ touch the circle at the points P & Q and the parabola at the points R & S. Then, the area (in sq units) of quadrilateral PQRS is ?

1. 3
2. 6
3. 9
4. 15

Q. The pair of tangents AP and AQ drawn from an external point to a circle with centre O are perpendicular to each other and length of each tangent is 5 cm. The radius of the circle is
(a) 10 cm (b) 7.5 cm (c) 5cm (d) 2.5 cm

Q. What is the distance between two parallel tangents of a circle of radius 4 cm?

Q.

If from any point on the common chord of two intersecting circles, tangents be drawn to the circles, prove that they are equal.

Q. In the given figure, O is the centre of a circle and PQ is a diameter. If $\angle ROS={40}^{\circ },$ then what is the value of $\angle RTS$?

1. ${20}^{\circ }$
2. ${45}^{\circ }$
3. ${70}^{\circ }$
4. ${65}^{\circ }$

Q. In the given figure, PT is a tangent to a circle with centre O. If OT = 6 cm and OP = 10 cm, then the length of tangent PT is
(a) 8 cm
(b) 10 cm
(c) 12 cm
(d) 16 cm

Q. PQ is a chord of length 16 cm of a circle of radius 10 cm. The tangent at P and Q intersect at a point T as shown in the figure. Find the length of TP [CBSE 2013C]

Q. In the given figure, O is the centre of the circle and TP is the tangent to the circle from an external point T. If ∠PBT = 30^∘ , prove that
BA : AT = 2 : 1 [CBSE 2015]

Q. In the given figure, PA and PB are two tangents from an external point P to a circle with centre O. If ∠PBA = 65^∘ , find ∠OAB and ∠APB

Q. In the given figure, PA and PB are two tangents to a circle with centre O, If ∠APB = 60^∘ then find the measure of ∠OAB

Q. In Fig. 10.85, PQ is a chord of a circle and PT is the tangent at P such that $\angle$QPT = 60⁰. Then , find $\angle$PRQ .

figure

Q.

In the given figure, PA is tangent to the circle. If PC intersects the circle at B and PB = 9 cm and BC =7cm, PA = ______ cm.

1. 18 cm

2. 12 cm

3. 16 cm

4. 10 cm

Q.

Two tangents to the circle $x^2+y^2=4$ at the points $A$and $B$ meet at $P(-4,0).$

The area of the quadrilateral $PAOB$ where $O$ is the origin, is

1. $4$ sq. units

2. $6\sqrt{2}$ sq. units

3. $4\sqrt{3}$ sq. units

4. None of these

Q.

Among the complex number z satisfying condition $|z+1-i|\le 1,$the number having the least positive argument is

1. $1-i$

2. $-1+i$

3. $-i$

4. None of these

Q. In the given figure, two circles touch each other at C and AB is a tangent to both the circles. The measure of ∠ACB is
(a) 45^∘
(b) 60^∘
(c) 90^∘
(d) 120^∘

Q.

AB and BC are the tangents to the circle from the points B. D is the centre of the circle. BD = 5 cm and CD = 3 cm. Find the value of AB - BC.

1. 20 cm

2. 4 cm

3. 0 cm

4. 9 cm

Q. If two tangents inclined at an angle of 60⁰ are drawn to a circle of radius 3 cm , then length of each tangent is equal to
(a) $\frac{3\sqrt{3}}{2}$ (b) 6 cm (c) 3 cm (d) $3\sqrt{3}$ cm

Q.

In quadrilateral ABCD; angle D = ${90}^o$, BC = 38 cm and DC = 25 cm. A circle is inscribed in this quadrilateral which touches AB at point Q such that QB = 27 cm. Finthe the radius of the circle.

Q.

In the above given figure, PB = 9 cm, CP = 4 cm and TP is a tangent at T. Find PT.

1. 2 cm

2. 6 cm

3. 9 cm

4. 8 cm

Q.

AB is a tangent to a circle with centre O and A is the point of contact. If , prove that AB = OA.

Q.

In the given figure 'O' is the center of the circle. SP and TP are the two tangents at S and T respectively. $\angle$SPT is ${50}^{\circ }$, the value of $\angle$SQT is:

1. 125°

2. 65°

3. 115°

4. None of the above

Q. In the given figure, O is the centre of two concentric circles of radii 6 cm and 10 cm. AB is a chord of outer circle which touches the inner circle. The length of AB is
(a) 8 cm
(b) 14 cm
(c) 16 cm
(d) $\sqrt{136}$ cm

Q.

In the given figure, 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.

In the picture below, the radius of the circle is 15 centimetres. Compute the lengths of the tangents PQ and PR

(i) The point P is equidistant from A and B

(ii) The line OP bisects the line AB and the angle APB

Q. In the given figure, PQ is a tangent to a circle with centre O. A is the point of contact. If ∠PAB = 67^∘, then the measure of ∠AQB is
(a) 73^∘
(b) 64^∘
(c) 53^∘
(d) 44^∘

Q. In the given figure, a circle is inscribed in a quadrilateral ABCD touching its sides Ab, BC, CD and AD at P, Q, R and S respectively.
If the radius of the circle is 10 cm, BC = 38 cm, PB = 27 cm and AD ⊥ CD then the length of CD is [CBSE 2013]
(a) 11 cm
(b) 15 cm
(c) 20 cm
(d) 21 cm

Q. If TP and TQ are 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.

In the circle centred at O, the tangents at A and B intersect at P. Prove the following:

(i)

the point P is equidistant from A and B

(ii)

the line OP bisects the line AB and the angle APB

(iii)

if the line OP cuts the line AB at Q, then OQ × OP = r², where r is the radius of the circle

Q. If from any point on the common chord of two interesting circle, tangents be drawn to the circles, prove that they are equal.

Q. In the given figure, O is the centre of a circle PT and PQ are tangents to the circle from an external point P. If ∠TPQ = 70^∘ then ∠TRQ
[CBSE 2015]