The circle C 1 : x 2 + y 2 = 3 , with centre at O , intersects the parabola x 2 = 2 y at the point P in the first quadrant. Let the tangent to the circle C 1 at P touches other two circles C 2 and C 3 at R 2 and R 3 , respectively. Suppose C 2 and C 3 have equal radii 2 3 and centres Q 2 and Q 3 , respectively. If Q 2 and Q 3 lie on the y -axis, then

area of the triangle O R 2 R 3 is 6 2

area of the triangle P Q 2 Q 3 is 4 2

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A circle $S$ passes through the point $(0, 1)$ and is orthogonal to the circles $(x - 1)^{2} + y^{2} = 16$ and $x^{2} + y^{2} = 1$ . Then,

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Important Questions on Circle

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Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-3>Q 11

The circle

C_{1} : x^{2} + y^{2} = 3

, with centre at

O

, intersects the parabola

x^{2} = 2 y

at the point

P

in the first quadrant. Let the tangent to the circle

C_{1}

P

touches other two circles

C_{2}

and

C_{3}

R_{2}

and

R_{3}

, respectively. Suppose

C_{2}

and

C_{3}

have equal radii

2 \sqrt{3}

and centres

Q_{2}

and

Q_{3}

, respectively. If

Q_{2}

and

Q_{3}

lie on the

y

-axis, then

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Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-3>Q 12

Let

R S

be the diameter of the circle

x^{2} + y^{2} = 1,

where

S

is the point

(1, 0)

. Let

P

be a variable point (other than

R

and

S

, on the circle and tangents to the circle at

S

and

P

meet at the point

Q

. The normal to the circle at

P

intersects a line drawn through

Q

parallel to

R S

at point

E

.Then, the locus of

E

passes through the point(s)

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Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-3>Q 14

Let

E_{1} E_{2}

and

F_{1} F_{2}

be the chords of circle

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

passing through the point

P_{0} (1, 1)

and parallel to the

X

-axis and the

Y

-axis, respectively. Let

G_{1} G_{2}

be the chord of

S

passing through

P_{0}

and having slope

- 1

. Let the tangents to

S

E_{1}

and

E_{2}

meet at

E_{3},

the tangents to

S

F_{1}

and

F_{2}

meet at

F_{3},

and the tangents to

S

G_{1}

and

G_{2}

meet at

G_{3} .

Then, the points

E_{3}, F_{3}

and

G_{3}

lie on the curve

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Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-3>Q 15

Let

P

be a point on the circle

S \equiv x^{2} + y^{2} = 4

with both coordinates being positive. Let the tangent to

S

P

intersect the coordinate axes at the points

M

and

N

. Then, the mid-point of the line segment

M N

must lie on the curve

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Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-3>Q 16

Let

T

be the line passing through the points

P (- 2, 7)

and

Q (2, - 5)

. Let

F_{1}

be the set of all pairs of circles

(S_{1}, S_{2})

such that

T

is tangents to

S_{1}

P

and tangent to

S_{2}

Q

, and also such that

S_{1}

and

S_{2}

touch each other at a point, say,

M

. Let

E_{1}

be the set representing the locus of

M

as the pair

(S_{1}, S_{2})

varies in

F_{1}

. Let the set of all straight line segments joining a pair of distinct points of

E_{1}

and passing through the point

R (1,1)

F_{2}

. Let

E_{2}

be the set of the mid-points of the line segments in the set

F_{2}

. Then, which of the following statement(s) is (are) TRUE?

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Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-3>Q 17

P

and

Q

are the points of intersection of the circles

x^{2} + y^{2} + 3 x + 7 y + 2 p - 5 = 0

and

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

, then there is a circle passing through

P, Q

and

(1, 1)

for

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Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-3>Q 18

The circle

x^{2} + y^{2} = 4 x + 8 y + 5

intersects the line

3 x - 4 y = m

at two distinct points if

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Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-3>Q 19

The two circles

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

and

x^{2} + y^{2} = c^{2} (c > 0)

touch each other, if

A circle S passes through the point (0,1) and is orthogonal to the circles (x-1)2+y2=16 and x2+y2=1. Then,

Important Questions on Circle

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A circle $S$ passes through the point $(0, 1)$ and is orthogonal to the circles $(x - 1)^{2} + y^{2} = 16$ and $x^{2} + y^{2} = 1$ . Then,