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There are two circles touch each other externally. Radius of the first circle with centre O is 8 cm. Radius of the second circle with centre A is 4 cm. Find the value of $k$ if the length of their common tangent BC is $k \sqrt{2}$ cm .

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Important Questions on Circle

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

The length of the transverse common tangent of the circles

x^{2} + y^{2} - 2 x + 4 y + 4 = 0

and

x^{2} + y^{2} + 4 x - 2 y + 1 = 0

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

Let the point

B

be the reflection of the point

A (2, 3)

with respect to the line

8 x - 6 y - 23 = 0 .

Let

T_{A}

and

T_{B}

be circle of radii

2

and

1

with centres

A

and

B

respectively. Let

T

be a common tangent to the circle

T_{A}

and

T_{B}

such that both the circle are on the same side of

T .

C

is the point of intersection of

T

and the line passing through

A

and

B,

then the length of the line segment

A C

is ___________.

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

Two straight lines

A B

and

A C

include an angle. A circle is drawn in this angle which touches both these lines. One more circle is drawn which touches both these lines as well as the previous circle. If the area of the bigger circle is

9

times the area of the smaller circle, then what must be the angle

A

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

If a circle

C

passing through the point

(4, 0)

touches the circle

x^{2} + y^{2} + 4 x - 6 y = 12

externally at the point

(1, - 1),

then the radius of

C

is:

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

Find the direct common tangents of the circles

x^{2} + y^{2} + 22 x - 4 y - 100 = 0 and x^{2} + y^{2} - 22 x + 4 y + 100 = 0

EASY

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

If the circles

C_{1} : x^{2} + y^{2} + 2 x + 4 y - 20 = 0, C_{2} : x^{2} + y^{2} + 6 x - 8 y + 9 = 0

have

n

common tangents and the length of the tangent drawn from the centre of similitude to the circle

C_{2}

l

then

\frac{l}{n^{2}} =

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

Show that

x^{2} + y^{2} - 6 x - 9 y + 13 = 0, x^{2} + y^{2} - 2 x - 16 y = 0

touch each other. Find the point of contact and the equation of common tangent at their point of contact.

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

If the incentre of an equilateral triangle is

(1, 1)

and the equation of its one side is

3 x + 4 y + 3 = 0

, then the equation of the circumcircle of this triangle is:

EASY

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

The distance between the centres of two circles having radii

5 c m

and

6 c m

\sqrt{10} c m

. Find the length (in

c m

) of direct common tangent.

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

Show that the circles

x^{2} + y^{2} - 6 x - 2 y + 1 = 0, x^{2} + y^{2} + 2 x - 8 y + 13 = 0

touch each other. Find the point of contact and the equation of common tangent at their point of contact.

EASY

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

Let

C_{1}

and

C_{2}

be the centres of the circles

x^{2} + y^{2} - 2 x - 2 y - 2 = 0

and

x^{2} + y^{2} - 6 x - 6 y + 14 = 0

respectively. If

P

and

Q

are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral

P C_{1} Q C_{2}

is :

EASY

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

Two circles of radii

8 cm

and

6 cm

touch each other externally. The length of the direct common tangent is:

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

If the area of an equilateral triangle inscribed in the circle

x^{2} + y^{2} + 10 x + 12 y + c = 0

27 \sqrt{3} sq . units,

then

c

is equal to:

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

The equation of the transverse common tangent of the circles

x^{2} + y^{2} - 6 x - 8 y + 9 = 0

and

x^{2} + y^{2} + 2 x - 2 y + 1 = 0

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

The circle passing through

(1, - 2)

and touching the axis of

x

(3, 0)

also passes through the point

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

The equation of a circle passing through the point

(1, 1)

and the point of intersection of the circles

x^{2} + y^{2} + 13 x - 3 y = 0

and

2 x^{2} + 2 y^{2} + 4 x - 7 y - 25 = 0

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

If a circle passing through the point

(- 1, 0)

touches

y

-axis at

(0, 2)

, then the

x

-intercept of the circle is

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

Show that the circles

x^{2} + y^{2} - 6 x - 9 y + 13 = 0, x^{2} + y^{2} - 2 x - 16 y = 0

touch each other. Find the point of contact and the equation of common tangent at their point of contact.

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

A point that lies on the common tangent to the circles

x^{2} + y^{2} - 2 x + 18 y + 78 = 0

and

x^{2} + y^{2} + 8 x - 6 y - 200 = 0

among the following options is

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

The common tangent to the circles

x^{2} + y^{2} = 4

and

x^{2} + y^{2} + 6 x + 8 y - 24 = 0

also passes through the point:

There are two circles touch each other externally. Radius of the first circle with centre O is 8 cm. Radius of the second circle with centre A is 4 cm. Find the value of k if the length of their common tangent BC is k2 cm .

Important Questions on Circle

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There are two circles touch each other externally. Radius of the first circle with centre O is 8 cm. Radius of the second circle with centre A is 4 cm. Find the value of $k$ if the length of their common tangent BC is $k \sqrt{2}$ cm .