Two Circles Touching Internally and Externally

Q.

Two circles of radii 18 cm and 8 cm touch externally. Find the length of a direct common tangent to the two circles.

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Q. In figure, a triangle ABC is drawn to circumscribe a circle of radius 3 cm, such that the segments BD and DC are respectively of lengths 6 cm and 9 cm.If the area of $\Delta ABC$ is 54 $c{m}^2$, then find the lengths of sides AB and AC.

Q.

If two circles intersect each other internally or externally, then in how many common tangents can be drawn?

1. 1

2. infinite

3. 2

4. 0

Q.

The centre of a circle$S=0$lies on$2x-2y+9=0$and $S=0$cuts orthogonally the circle $x^2+y^2=4.$ Then the circle must pass through the point

1. $(1,1)$

2. $(-\frac{1}{2},\frac{1}{2})$

3. $(5,5)$

4. $(-4,4)$

Q. Theorem : The chords corresponding to congruent arcs of a circle (or congruent circles) are congruent.

Q.

Find the point on the hyperbola $\mathrm{xy}=8$ that is closest to the point $(3,0)$.

Q.

What is the maximum and minimum distance of the point (9, 12) from the circle $x^2+y^2-6x-8y-24=0$

1. 15 and 5

2. 14 and 6

3. 17 and 3

4. 12 and 8

Q.

Three circle touch each other externally. A triangle is formed when the centres of these circles are joined together. Find the radii of the circles, if the sides of the triangle formed are 6 cm, 8 cm and 9 cm

1. $r_2=5.5cm$
   $r_3=3.5cm$
   $r_1=3.5cm$

2. $r_2=5.9cm$
   $r_3=3.5cm$
   $r_1=2.5cm$

3. $r_2=5.9cm$
   $r_3=1.5cm$
   $r_1=2.5cm$

4. $r_2=5.9cm$
   $r_3=9.5cm$
   $r_1=2.5cm$

Q.

Two circle touch each externally at point P. Q is a point on the common tangent through P. Then, the tangents QA and QB are equal.

1. False

2. True

Q.

If the tangents PA and PB from an external point P to circle with centre O are inclined to each other at an angle of ${40}^0$, then $\angle$POA =

1. 70°

2. 80°

3. 50°

4. 60°

Q. In the given figure, a circle touches all the four sides of a quadrilateral ABCD with AB = 12cm, BC = 10 cm and CD = 14cm. Find AD.

1. 10
2. 12
3. 15
4. 16

Q. In the given figure, PQ and PR are two tangents drawn from an external point P to a circle with centre 'O'. If OQ = 3 cm and PS = 2 cm then, find the perimeter (in cm) of quadrilateral PQOR.

1. 14

Q. In the given figure, PQ and PR are two tangents drawn from an external point P to a circle with centre 'O'. If OQ = 3 cm and PS = 2 cm then, find the perimeter (in cm) of quadrilateral PQOR.

1. 14

Q. A circle is inscribed in a $△ABC$ touching $AB,BC$ and $AC$ at $P,Q$ and $R$ respectively. If $AB=10cm$ $AR=7cm$ and $CR=5cm$, then find the length of $BC$

Q.

Find the distance between the centers of the circles in the figure.

1. 14cm

2. 16cm

3. 13cm

4. 19cm

Q. In the adjoining figure, the circles with centres $P$ and $Q$ touch each other at $RA$ line passing through $R$ meets the circles at $A$ and $B$ respectively. Prove that -
i. seg AP $\parallel$ seg $BQ$,
ii. $△APR\sim △RQB$, and
iii. Find $\angle RQB$ if $\angle PAR={35}^{\circ }$.

Q.

Dylan is organizing a curling tournament. The sports complex charges Rs. 115/ h for the ice rental. Dylan has booked it for 6h. He will charge each of the 14 teams in the tournament an entrance fee. How much should he charge from each team in order to make a profit?

1. x < Rs. 23

2. x ≥ Rs. 50

3. x > Rs. 43

4. x < Rs. 120

Q.

State True or False:
O is any point on the bisector of the acute angle $\angle$XYZ. The line OP$\parallel$ZY. Then, $△$YPO is an Isosceles right-angled triangle.

1. Scalene

2. Isosceles but not right angled

Q. Let $R$ be the region of the disc $x^2+y^2\le 1$ in the first quadrant. Then the area of the largest possible circle contained in $R$ is

1. $\pi (4-3\sqrt{2})$
2. $\pi (3-2\sqrt{2})$
3. $\pi (2\sqrt{2}-2)$
4. $\frac{\pi }{6}$

Q. Prove that the circle $x^2+y^2+2ax+c^2=0$ and $x^2+y^2+2by+c^2=0$ touches each other, if $\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{c^2}$.

Q. Examine if the two circles $x^2+y^2-2x-4y=0$ and $x^2+y^2-8y-4=0$ touch each other externally or internally

Q.

With the vertices of $△ABC$ as centers, three circles are described, each touching the other two externally. If the sides of the triangle are 9 cm, 7 cm and 6 cm, find the radii of the circles. [4 MARKS]

Q. In a right angled triangle $BC=5,AB=4,AC=3$. Let $S$ be the circum circle. Let $S_1$ be the circle touching both sides $AB$ and $AC$ and circle $S$ internally. Let $S_2$ be the circle touching the sides $AB$ and $AC$ of $△ABC$ and touching the circle $S$ externally. If $r_1{r}_2$ are radii of circles $S_1$ and $S_2$ respectively then $r_1{r}_2=$

1. 12
2. 24
3. 15
4. 42

Q. If a circle be drawn so as always to touch a given straight line and also a given circle externally then prove that the locus of its centre is a parabola. (given line and given circle are non intersecting)

Q.

1. They touch internally each other
2. One is inside another
3. They touch externally each other
4. They are separated

Q. The two circles $x^2+y^2-2x+6y+6=0$ and $x^2+y^2-5x+6y+15=0$ touch each other

1. Externally
2. Internally
3. Coincide
4. None of these

Q. Three circle of radius 3, 4, 5 touch each other externally. Tangents are drawn at the point of contact, which meet at P. Find the distance between point of contact and P.

Q. A line intersecting a circle in two points is called

1. tangent
2. secant
3. point of contact
4. None of these

Q. If two circles with radii $8cm$ and $3cm$ respectively touch internally, then find the distance between their centres.

Q. Two circles with radii $4cm$ and $3cm$ touch each other externally. Find the distance between their centres.