Centre of a Circle Lies on the Bisector of Angle between Two Tangents

Q.

$△ABC$ is a right-angled triangle right angled at$\mathrm{B}$ such that$\mathrm{BC}=6\mathrm{cm}$ and $\mathrm{AB}=8\mathrm{cm}$. A circle with center $O$ is inscribed in triangle $△ABC$. The radius of the circle is.

1. $1cm$

2. $2cm$

3. $3cm$

4. $4cm$

Q. Question 6
A circular park of radius 20 m is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk each other. Find the length of the string of each phone.

Q.
In the figure given, the radius of the circle is 15cm. The angle subtended by the chord AB at the centre O is ${60}^{\circ }$. Find the area of the Major segment.
1. $588.75c{m}^2$
2. $117.75c{m}^2$
3. $686.05c{m}^2$
4. $97.3c{m}^2$

Q.

In the diagram the smaller circle is drawn such that it passes through the Centre of the larger and touches the larger circle. What is the radius of the smaller circle if the diameter of the larger circle is D?

1. D/2

2. D

3. 2D

4. D/4

Q.

In the given figure, if the angle between two radii of a circle is ${120}^{\circ }$, then find $\angle$PAO.

1. ${90}^{\circ }$
2. ${45}^{\circ }$
3. ${30}^{\circ }$
4. ${60}^{\circ }$

Q. In the given figure, $XY$ and $X^1$$Y^1$ are two parallel tangents to a circle with centre O and another tangent $AB$ with point of contact C intersecting $XY$ at $A$ and $X^1$$Y^1$ at $B$ then $\angle AOB$ is ___.

1. ${100}^{\circ }$
2. ${60}^{\circ }$
3. ${80}^{\circ }$
4. ${90}^{\circ }$

Q. Two tangents AP and AQ are drawn from an external point A to a circle with centre 'O'. If tangents AP and AQ are inclined to each other at an angle of ${60}^{\circ }$ then, $\angle$POA is ____.

1. ${120}^{\circ }$
2. ${60}^{\circ }$
3. ${70}^{\circ }$
4. ${90}^{\circ }$

Q. In the given figure, AP and AQ are two tangent drawn from a point outside the circle. Select the correct option from the following.

1. $\angle$PAO > $\angle$QAO
2. $\angle$PAO = 2 $\angle$QAO
3. $\angle$PAO < $\angle$QAO
4. $\angle$PAO = $\angle$QAO

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 intersects XY at A and X'Y' at B. Then $\angle$ AOB is _____.

1. ${80}^{\circ }$
2. ${60}^{\circ }$
3. ${85}^{\circ }$
4. ${90}^{\circ }$

Q. In figure 1, PQ is a chord of a circle with centre O and PT is a tangent. If $\angle QPT={60}^{\circ }$, find $\angle PRQ$.

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-

1. $7$ cm
2. $12$ cm
3. $15$ cm
4. $24.5$ cm

Q. In the given figure, $XY$ and $X^1$$Y^1$ are two parallel tangents to a circle with centre O and another tangent $AB$ with point of contact C intersecting $XY$ at $A$ and $X^1$$Y^1$ at $B$ then $\angle AOB$ is ___.

1. ${100}^{\circ }$
2. ${80}^{\circ }$
3. ${90}^{\circ }$
4. ${60}^{\circ }$

Q. Let these base $AB$ of a triangle $ABC$ be fixed and the vertex $C$ lie on a fixed circle of radius $r$. Lines through $A$ and $B$ are drawn to intersect $CB$ and $CA$, respectively, at $E$ and $F$ such that $CE:EB=1:1$ and $CF:FA=1:2$. If the point of intersection $P$ of these lines lies on the median through $AB$ for all positions of $AB$ then the focus of $P$ is

1. a circle of radius $\frac{r}{2}$
2. a circle of radius $2r$
3. a parabola of latus rectum $4r$
4. a rectangular hyperbola

Q. In the given figure, PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length of TP.

1. 6.67 cm
2. 8.2 cm
3. 9.3 cm
4. 7.8 cm

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 intersects XY at A and X'Y' at B. Then $\angle$ AOB is _____.

1. ${80}^{\circ }$
2. ${90}^{\circ }$
3. ${85}^{\circ }$
4. ${60}^{\circ }$