HARD

JEE Main

IMPORTANT

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The diameter $A B$ of a circle of radius $2$ is extended to a point $D$ outside the circle so that $B D = 3 .$ Point $E$ is chosen so that $E D = 5$ and the line $E D$ is perpendicular to the line $A D$ . Segment $A F$ intersects the circle at point $C$ between $A$ and $E .$ The area of $∆ A B C$ is $\frac{P}{S}$ , where $P$ and $S$ are co-prime. What is $P - 2 S ?$

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Important Questions on Circle

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 4 - Circle>Level 3>Q 6

Let

C D

be a chord of a circle

Γ_{1}

and

A B

a diameter of

Γ_{1}

perpendicular to

C D

N

with

A N > N B .

A circle

Γ_{2}

centred at

C

with radius

C N

intersects

Γ_{1}

at points

P

and

Q,

and the segments

P Q

and

C D

intersect at

M .

Given that the radii of

Γ_{1}

and

Γ_{2}

are

61

units and

60

units respectively, find the length of

A M .

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 4 - Circle>Level 3>Q 7

If circular arcs $A C$ and $B C$ have centres at $B$ and $A$ , respectively, then there exists a circle tangent to both $\hat{A C}$ and $\hat{B C}$ , and to $\bar{A B}$ . If the length of $\hat{B C}$ is $12$ , then the circumference of the circle is

Question Image

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 4 - Circle>Level 3>Q 8

A line meets the co-ordinate axes in

A

and

B .

A circle is circumscribed about the triangle

O A B

. If

d_{1}

and

d_{2}

are the distances of the tangent to the circle at the origin

O

from the points

A

and

B

, respectively, then the diameter of the circle is

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 4 - Circle>Level 3>Q 9

Let the tangent to the circle

x^{2} + y^{2} = 25

at the point

R (3, 4)

meet

x

-axis and

y

-axis at point

P

and

Q

, respectively. If

r

is the radius of the circle passing through the origin

O

and having centre at the incentre of the triangle

O P Q,

then

r^{2}

is equal to

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 4 - Circle>Level 3>Q 10

The locus of the point of intersection of tangents to the circle

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

which are inclined at an angle

α

with each other, is

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 4 - Circle>Level 3>Q 11

If the circle

x^{2} + y^{2} + 2 g x + 2 f y + c = 0

is touched by the line

y = x

at point

P

such that

O P = 6 \sqrt{2}

units, where

O

is the origin, then the value of

c

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 4 - Circle>Level 3>Q 12

Let

|(n - a)! - t| + |t - (n - b)!| + |a + b - k_{1} n - k_{2}| \leq 0 \forall n

such that

a < b < n

and

a, b, n, t \in N & k_{1}, k_{2} \in I

. If

P, Q

be any two points on the curve

y = \log_{\frac{1}{2}} (x + \frac{k_{2}}{2}) + \log_{2} (\sqrt{4 x^{2} + 4 k_{2} x + (k_{1} + k_{2})})

. Also point

P

lies on

x^{2} + y^{2} = k_{1}^{3} - 2 k_{2}

and point

Q

lies inside the given circle such that its abscissa is an integer. Then the minimum value of

\vec{O P} . \vec{O Q}

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 4 - Circle>Level 3>Q 13

The point

P

moves in the plane of a regular hexagon, such that the sum of squares of its distances from the vertices of hexagon is

6 a^{2} .

If the radius of the circumcircle of hexagon is

r (< a),

then the locus of

P

Important Questions on Circle

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