Touching Circles Theorem

Q.

Find the coordinates of the centre of the circle passing through the points (0, 0), (–2, 1) and (–3, 2). Also, find its radius.

Q.

In the given figure, two circles touch each other externally at point $P$. $AB$ is the direct common tangent of these circles. Prove that :

(i) tangent at point $P$ bisects $AB$,

(ii) angle $APB=$ ${90}^{\circ }$.

Q.

In the given figure, there are two circles with the centres O and O' touching each other internally at P. Tangents TQ and TP are drawn to the larger circle and tangents TP and TR are drawn to the smaller circle . Find TQ : TR

1. 8 : 7

2. 7 : 8

3. 5 : 4

4. 1 : 1

Q. Prove that: If two circles touch each other, then the point of contact will lie on the line joining the two centres.

Q.

Two circles touch each other internally at a point P. A chord AB of the bigger circle intersects the other circle in C and D. Prove that : $\angle CPA=\angle DPB$.

Q.

The angle A of the triangle ABC is a right angle. The circle with AC as diameter cuts BC at point D. If BD = 6 cm & DC = 18 cm , calculate the length of AB.

1. 9 cm

2. 12 cm

3. 8 cm

4. 63 cm

Q.

In the given figure, O is the center of the circumcircle of triangle ABC. Tangents at A and B intersect at T. If $\angle ATB={80}^{\circ }$ and $\angle AOC={130}^{\circ }$, Calculate $\angle CAB$. [4 MARKS]

Q.

Prove that the lengths of tangents drawn from an external point to a circle are equal. Using the above do the following:

ABC is an isosceles triangle in which AB = AC, circumscribed about a circle as shown in the fig. Prove that the base is bisected by the point of contact.

Q.

ABC is a triangle with AB = 10 cm, BC = 8 cm and AC = 6 cm. Three circles are drawn touching each other with the vertices as their centres. Find the sum of the radii of the three circles.

1. 7 cm

2. 3 cm

3. 25 cm

4. 12 cm

Q.

Two circles intersect each other at $A$ and $B$. The common chord $AB$ is produced to meet common tangent $PQ$ to the circle at $D$. Prove that $\mathrm{PQ}$ is bisected at $D$.

Q.

Radii of 2 circles are 16 cm and 8 cm. Find the distance between their centres if they touch internally.

1. 8 cm

2. 2 cm

3. 4 cm

4. 24 cm

Q.

Radii of 2 circles are 6 cm and 8 cm. Find the distance between their centres if they touch internally.

1. 8 cm

2. 2 cm

3. 4 cm

4. 24 cm

Q. Question 7
If figure common tangents AB and CD to two circles intersect at E. Prove that AB = CD.

Q. Question 1
State whether the following is true or false.
By geometrical construction, it ispossible to divide a line segment in the ratio $\sqrt{3}:\frac{1}{\sqrt{3}}$.

Q.

A square is inscribed in a circle. Calculate the ratio of the area of the circle and the square.

Q. Given a circle with centre O and OB is perpendicular to the chord AM. If the length of AM is 8 cm, then find the length of AB.

1. 5 cm
2. 4 cm
3. 6 cm
4. 2 cm

Q. In the given figure, common tangents AB and CD to the two circle with centres O₁ and O₂ intersect at E. Prove that AB = CD [CBSE 2014]

Q. Two circles $x^2+y^2+2ax+c=0$ and $x^2+y^2+2by=0$ touch if

1. $a^2+b^2=c^2$
2. $\frac{1}{c^2}=\frac{1}{-a^2}+\frac{1}{-b^2}$
3. $\frac{1}{c^2}=\frac{1}{a^2}+\frac{1}{b^2}$
4. $\frac{1}{c}=\frac{1}{a^2}+\frac{1}{b^2}$

Q.

The distance between the tops of two trees $20m$ and $28m$ high is $17m.$ The horizontal distance between the feet of the trees is

1. 15 m

2. 9 m

3. 11 m

4. 31 m

Q. The line $\frac{1}{r}=A\cos \theta +B\sin \theta ,$ touches the circle $r=2a\cos \theta ,$ then

1. $a^2{B}^2+2aA=1$
2. $a^2{A}^2+2aB=1$
3. $a^2{A}^2+2aA=1$
4. $a^2{B}^2-2aA=1$

Q. Question 13
Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the center of the circle.

Q. If radii of two circles are 4 cm and 2.8 cm. Find distance between their centres and also draw figure of these circles touching each other - (i)externally (ii)internally
$\left[2Marks\right]$

Q. Two circles intersect each other at A and B.Let DC be a common tangent touching the circle in point C and D
Prove that $\angle \mathrm{CAD}+\angle \mathrm{CBD}=180°$

Q. Show that the circles
$x^2+y^2-10x-4y-20=0$
and $x^2+y^2+14x-6y+22=0$
touch each other , Find the co- ordinates of the point of contact and the equation of the common tangent at the point of contact.

Q.

Two tangents are drawn onto a circle with centre O and radius r, from an external point P. These tangents touch the circle at points A and B respectively. The triangle ABP is definitely a ______________ .

1. Isosceles triangle

2. Equilateral triangle

3. Scalene triangle

4. Right angled triangle

Q. Two circles $x^2+y^2-2kx=0$ and $x^2+y^2-4x-8y+16=0$ touch each other externally. Then, $k$ is

1. $4$
2. $1$
3. $2$
4. $-4$

Q. ABCD is a square, in which a circle is inscribed touching all the sides of square. In the four corners of square, 4 smaller circles of equal radii are drawn, containing maximum possible area.
What is the ratio of the area of larger circle to that of sum of the areas of four smaller circles?

Q.

AP = x, PC = y and OP = r, the in radius of the right angled triangle ABC, with B right angled.

Find the value of (1 + $\frac{r}{x}$)(1 + $\frac{r}{y}$)

Q. Equation of the circle which touches the lines $x=0,y=0$ and $3x+4y=4$ is

1. $x^2-4x+y^2+4y+4=0$
2. $x^2-4x+y^2+4y-4=0$
3. $x^2+4x+y^2+4y+4=0$
4. $x^2-4x+y^2-4y+4=0$

Q. A circle of radius $2$ unit lies in the first quadrant and touches both the axes of coordinates, equation of circle with center at $(6,5)$ and touching the above circle externally is

1. $(x-6{)}^2+(y-5{)}^2=9$
2. $x^2+y^2-10x-12y+52=0$
3. $x^2+y^2-12x-10y+32=0$
4. $x^2+y^2+12x-10y-32=0$