Tangents from a Given Point

Q. In the given figure, XY and X'Y' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C, is intersecting XY at A and X'Y' at B. Prove that
$\angle$ AOB = 90${}^{\circ }$.

Q.

In fig., CP and CQ are tangents from an external point C to a circle with center O. AB is another tangent which touches the circle at R. If CP = 11 cm and BR = 4 cm, find the length of BC

Q. O is a point outside the circle and OA and OB are the two tangents drawn from the external point 'O'. If OA = 10cm, find the length of OB.

1. 10 cm
2. 8 cm
3. 5 cm
4. 20 cm

Q.

Two circles touch each other externally at a point C and P is a point on the common tangent at C. If PA and PB are tangents to the two circles, prove that PA = PB.

Q. The number of common tangents to two circles which are touching each other internally is 2.

1. True
2. False

Q. In the given below figure, from an external point P, a tangent PT and a secant PAB is drawn to a
circle with centre O. ON is perpendicular on the chord AB. Prove that PA. PB = PT2.

Q.

In the figure, AP and AQ are tangents to the circle with centre O, from an external point A. If ∠PAQ = 70^o then APQ is equal to:

1. 35^o

2. 125^o

3. 110^o

4. 55^o

Q.

The centres of two circles of radii $3cm$ and $2cm$ are $8cm$ apart. Find the length of the common tangent.

1. 39

2. $\sqrt{39}$

3. $64$

4. $25$

Q.

In the figure below, ABC is a triangle, AC and BC are tangents to the circle of center O at two different points. The lengths of AM, AR and BC are equal to 6 cm, 12 cm and 18 cm respectively. Find the radius of the circle.

Q.

In the given figure, two circles touch each other externally at point, P. AB is the direct common tangent of these circles. Select the statements that are true.

1. Tangent at point P bisects AB.

2. $\angle APB={40}^{\circ }$

3. $\angle APB={45}^{\circ }$

4. $\angle APB={10}^{\circ }$

Q.

Maximum number of tangents parallel to a given secant of a circle is/ are

1. 2

2. 1

3. infinite

4. no tangents are possible

Q. The number of common tangents to two circles which are intersecting each other is 2.

1. True
2. False

Q.

In the figure given below, PA and PB are tangents to the circle from point P. When the centre of the circle O and the point P are joined, how are the values of $x$ and $y$ related?

1. $x=y$

2. $x+y={90}^{\circ }$

3. $x>y$

4. $x<y$

Q.

If two circles touches each other internally then they have ____ common tangent/tangents.

1. 1

2. 2

3. 3

4. Infinite

Q. In an online test, Ishita comes across the statement - If a tangent is drawn to a circle from an external point, then the square of length of tangent drawn is equal to difference of squares of distance of the tangent from the centre of circle and radius of the circle.Help Ishita, in answering the following questions based on the above statement.
(i) If $AB$ is a tangent to a circle with centre $O$ at $B$ such that $AB=10cm$ and $OB=5cm$, then $OA=$
(a) $3\sqrt{5}cm$
(b) $5\sqrt{5}cm$
(c) $4\sqrt{5}cm$
(d) $6\sqrt{5}cm$

(ii) In the adjoining figure, radius of the circle is

(a) $8cm$
(b) $7cm$
(c) $9cm$
(d) $10cm$

(iii)In the adjoining figure, length of tangent $AP$ is

(a) $12cm$
(b) $24cm$
(c) $30cm$
(d) None of these

(iv) $PT$ is a tangent to a circle with centre $O$ and diameter $=40cm$. If $PT=21cm$, then $OP=$
(a) $33cm$
(b) $29cm$
(c) $37cm$
(d) None of these

(v) In the adjoining figure, the length of the tangent is

(a) $15cm$
(b) $9cm$
(c) $8cm$
(d) $10cm$

Q. In the given figure, there are two circles that touch each other at A. P is a point which is at a distance of 25 cm from O as shown in the figure. The radius of the circle with centre O is 7 cm. The diameter of the circle with centre O’ is 10 cm. What is the length of the tangent PC?

1. 25 cm
2. 24 cm
3. 26 cm
4. 27 cm

Q. Two circles of radii $4cm$ and $2cm$ have their centres $10cm$ apart. Draw a two direct common tangents and measure their length.

Q. The maximum number of tangents that can be drawn to a circle from a point outside it is ______.
[1 Mark]

Q. From an external point P, two tangents are drawn that touches the circle at points Q and R. The triangle PQR is necessarily a/an ___triangle.

1. Equilateral
2. Scalene
3. Isosceles
4. Right angled

Q. In the given figure, there are two circles that touch each other at A. P is a point which is at a distance of 25 cm from O as shown in the figure. The radius of the circle with centre O is 7 cm The diameter of the circle with centre O’ is 10cm. What is the length of the tangent PC?

1. 25 cm
2. 24 cm
3. 26 cm
4. 27 cm

Q. If two circles touch each other externally, then they have 3 common tangents.

1. True
2. False

Q. The maximum number of tangents parallel to a given secant of a circle are four.

1. True
2. False

Q. From an external point P, two tangents are drawn that touches the circle at points Q and R. The triangle PQR is necessarily a/an ___triangle.

1. equilateral
2. isosceles
3. scalene
4. right angled

Q.

In the adjoining figure, if AD, AE and BC are tangents to the circle at D, E and F respectively. Then,

1. AD = AB + BC + CA

2. 2AD = AB + BC + CA

3. 3AD = AB + BC + CA

4. 4AD = AB + BC + CA

Q.

If TP and TQ are two tangents drawn to a circle with center O such that $\angle$POQ $={110}^{\circ }$, then, $\angle$PTQ is equal to:

1. 60°

2. 70°

3. 80°

4. 90°

Q. Prove that the line segment joining the points of contact of two parallel tangent of a circle, passes through its centre.

Q. There are two circles with centres $c_1\&c_2$ with having radii $3cm$. Distance between their centres is $8cm$
Length of transverse common tangent is ........... cm

1. $5$
2. $\sqrt{26}$
3. $3\sqrt{3}$
4. $\sqrt{28}$

Q.

In the figure given below, PA and PB are tangents to the circle from point P. When the centre of the circle O and the point P are joined, how are the values of x and y related?

1. $x>y$

2. $x<y$

3. $x=y$

4. $x+y={90}^{\circ }$

Q.

In the given figure, there are two circles that touch each other at A. P is a point which is at a distance of $25cm$ from O as shown in the figure. The radius of the circle with centre O is $7cm$. The diameter of the circle with centre O’ is $10cm.$ What is the length of the tangent PC?

1. $25cm$

2. $24cm$

3. $26cm$

4. $27cm$

Q.
In the figure given above, PA and PB are tangents to the circle from point P. How many real values of $x$ exist such that the length of PA is $x$ and length of PB is $x^2+1$?

1. 0
2. 1
3. 2
4. 3