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

HARD

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IMPORTANT

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The straight line $2 x - 3 y = 1$ divides the circular region $x^{2} + y^{2} \leq 6$ into two parts. If $S = \{(2, \frac{3}{4}), (\frac{5}{2}, \frac{3}{4}), (\frac{1}{4}, - \frac{1}{4}), (\frac{1}{8}, \frac{1}{4})\}$ , then the number of point(s) in $S$ lying inside the smaller part is ________.

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Important Questions on Circle

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-3>Q 7

The locus of the midpoint of the chord of contact of tangents drawn from points lying on the straight line

4 x - 5 y = 20

to the circle

x^{2} + y^{2} = 9

HARD

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-3>Q 9

Circle(s) touching the

x

-axis at a distance of

3

units from the origin and having an intercept of length

2 \sqrt{7}

on the

y

-axis is (are)

HARD

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-3>Q 10

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,

HARD

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IMPORTANT

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

HARD

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IMPORTANT

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)

HARD

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IMPORTANT

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

HARD

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IMPORTANT

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

HARD

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IMPORTANT

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?

The straight line 2x-3y=1 divides the circular region x2+y2≤6 into two parts. If S=2, 34, 52, 34, 14, -14, 18, 14, then the number of point(s) in S lying inside the smaller part is ________.

Important Questions on Circle

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The straight line $2 x - 3 y = 1$ divides the circular region $x^{2} + y^{2} \leq 6$ into two parts. If $S = \{(2, \frac{3}{4}), (\frac{5}{2}, \frac{3}{4}), (\frac{1}{4}, - \frac{1}{4}), (\frac{1}{8}, \frac{1}{4})\}$ , then the number of point(s) in $S$ lying inside the smaller part is ________.