Position of a Point with Respect to a Circle

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JEE Advanced 2008

LEVELJEE Main

Consider $L_{1} : 2 x + 3 y + p - 3 = 0$ , $L_{2} : 2 x + 3 y + p + 3 = 0$ where $p$ is a real number, and $C : x^{2} + y^{2} + 6 x - 10 y + 30 = 0$ . STATEMENT-1 : If line $L_{1}$ is a chord of circle $C$ , then line $L_{2}$ is not always a diameter of circle $C$ and STATEMENT-2 : If line $L_{1}$ is a diameter of circle $C$ , then line $L_{2}$ is not a chord of circle $C$ .

(A)

Statement - 1 is True, Statement - 2 is True; Statement - 2 is a correct explanation for Statement - 1

(B)

Statement - 1 is True, Statement - 2 is True; Statement - 2 is NOT a correct explanation for Statement - 1

(C)

Statement - 1 is True, Statement - 2 is False

(D)

Statement - 1 is False, Statement - 2 is True

JEE Advanced 2011

LEVELJEE Main

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

JEE Main 2019 (12 January Shift 1)

LEVELJEE Main

If a variable line, $3 x + 4 y - λ = 0$ is such that the two circles $x^{2} + y^{2} - 2 x - 2 y + 1 = 0$ and $x^{2} + y^{2} - 18 x - 2 y + 78 = 0$ are on its opposite sides, then the set of all values of $λ$ is the interval :-

(A)

[12, 21]

(B)

(2, 17)

(C)

(23, 31)

(D)

[13, 23]

JEE Main 2021 (20 July Shift 2)

LEVELJEE Main

Let $r_{1}$ and $r_{2}$ be the radii of the largest and smallest circles, respectively, which pass through the point $(- 4, 1)$ and having their centres on the circumference of the circle $x^{2} + y^{2} + 2 x + 4 y - 4 = 0$ . If $\frac{r _{1}}{r _{2}} = a + b 2$ , then $a + b$ is equal to :

JEE Main 2021 (22 July Shift 1)

LEVELJEE Main

Let the circle $S : 36 x^{2} + 36 y^{2} - 108 x + 120 y + C = 0$ be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, $x - 2 y = 4$ and $2 x - y = 5$ lies inside the circle $S$ , then :

(A)

\frac{25}{9} < C < \frac{13}{3}

(B)

100 < C < 165

(C)

81 < C < 156

(D)

100 < C < 156

JEE Main 2021 (17 March Shift 1)

LEVELJEE Main

Choose the incorrect statement about the two circles whose equations are given below: $x^{2} + y^{2} - 10 x - 10 y + 41 = 0$ and $x^{2} + y^{2} - 16 x - 10 y + 80 = 0$

(A)

Distance between two centres is the average of radii of both the circles.

(B)

Both circles' centres lie inside region of one another.

(C)

Both circles pass through the centre of each other.

(D)

Circles have two intersection points.

JEE Main 2022 (27 June Shift 2)

LEVELJEE Main

The set of values of $k$ for which the circle $C : 4 x^{2} + 4 y^{2} - 12 x + 8 y + k = 0$ lies inside the fourth quadrant and the point $(1, - \frac{1}{3})$ lies on or inside the circle $C$ is :

(A)

An empty set

(B)

(6, \frac{95}{9}]

(C)

[\frac{80}{9}, 10)

(D)

(9, \frac{92}{9}]

JEE Main 2024 (05 Apr Shift 2)

LEVELJEE Advanced

Let the maximum and minimum values of $(8 x - x^{2} - 12 - 4)^{2} + (x - 7)^{2}, x \in R$ be $M$ and $m$ , respectively. Then $M^{2} - m^{2}$ is equal to _________

JEE Advanced 2021

LEVELJEE Advanced

Comprehension Passage

Let

M = {(x, y) \in R \times R : x^{2} + y^{2} \leq r^{2}}

, where

r > 0

. Consider the geometric progression

a_{n} = \frac{1}{2 ^{n - 1}}, n = 1, 2, 3, \dots

. Let

S_{0} = 0

and, for

n \geq 1

, let

S_{n}

denote the sum of the first

n

terms of this progression. For

n \geq 1

, let

C_{n}

denote the circle with center

(S_{n - 1}, 0)

and radius

a_{n}

, and

D_{n}

denote the circle with center

(S_{n - 1}, S_{n - 1})

and radius

a_{n}

Question 1:

Consider $M$ with $r = \frac{1025}{513}$ . Let $k$ be the number of all those circles $C_{n}$ that are inside $M$ . Let $l$ be the maximum possible number of circles among these $k$ circles such that no two circles intersect. Then

(A)

k + 2 l = 22

(B)

2 k + l = 26

(C)

2 k + 3 l = 34

(D)

3 k + 2 l = 40

Question 2:

Consider $M$ with $r = \frac{( 2 ^{199} - 1 ) 2}{2 ^{198}}$ . The number of all those circles $D_{n}$ that are inside $M$ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Choose the incorrect statement about the two circles whose equations are given below: x2+y2−10x−10y+41=0 and x2+y2−16x−10y+80=0

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The set of values of k for which the circle C:4x2+4y2−12x+8y+k=0 lies inside the fourth quadrant and the point (1,−31​) lies on or inside the circle C is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the maximum and minimum values of (8x−x2−12​−4)2+(x−7)2,x∈R be M and m, respectively. Then M2−m2 is equal to _________

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Consider M with r=2198(2199−1)2​​. The number of all those circles Dn​ that are inside M is

Choose the incorrect statement about the two circles whose equations are given below: $x^{2} + y^{2} - 10 x - 10 y + 41 = 0$ and $x^{2} + y^{2} - 16 x - 10 y + 80 = 0$

The set of values of $k$ for which the circle $C : 4 x^{2} + 4 y^{2} - 12 x + 8 y + k = 0$ lies inside the fourth quadrant and the point $(1, - \frac{1}{3})$ lies on or inside the circle $C$ is :

Let the maximum and minimum values of $(8 x - x^{2} - 12 - 4)^{2} + (x - 7)^{2}, x \in R$ be $M$ and $m$ , respectively. Then $M^{2} - m^{2}$ is equal to _________

Consider $M$ with $r = \frac{( 2 ^{199} - 1 ) 2}{2 ^{198}}$ . The number of all those circles $D_{n}$ that are inside $M$ is