Circles · JEE Main previous year questions

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2023 30 questions

Q192023 · 13 Apr 2023 · Shift 1
Let the tangent and normal at the point $(33, 1)$ on the ellipse $\frac{x ^{2}}{36} + \frac{y ^{2}}{4} = 1$ meet the $y$ -axis at the points $A$ and $B$ respectively. Let the circle $C$ be drawn taking $A B$ as a diameter and the line $x = 25$ intersect $C$ at the points $P$ and $Q$ . If the tangents at the points $P$ and $Q$ on the circle intersect at the point $(α, β)$ , then $α^{2} - β^{2}$ is equal to :
1. A. $61$
2. B. $\frac{304}{5}$
3. C. $60$
4. D. $\frac{314}{5}$
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Answer: B
Q222023 · 13 Apr 2023 · Shift 1· Numerical answer
Let $w = z \overset{z}{ˉ} + k_{1} z + k_{2} i z + λ (1 + i), k_{1}, k_{2} \in R$ . Let $Re (w) = 0$ be the circle $C$ of radius 1 in the first quadrant touching the line $y = 1$ and the $y$ -axis. If the curve $Im (w) = 0$ intersects $C$ at A and B, then $30 (A B)^{2}$ is equal to _________
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Answer: 24
Q152023 · 13 Apr 2023 · Shift 2
Let the centre of a circle $C$ be $(α, β)$ and its radius $r < 8$ . Let $3 x + 4 y = 24$ and $3 x - 4 y = 32$ be two tangents and $4 x + 3 y = 1$ be a normal to $C$ . Then $(α - β + r)$ is equal to :
1. A. $7$
2. B. $9$
3. C. $5$
4. D. $6$
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Answer: A
Q62023 · 12 Apr 2023 · Shift 1
Let $C$ be the circle in the complex plane with centre $z_{0} = \frac{1}{2} (1 + 3 i)$ and radius $r = 1$ . Let $z_{1} = 1 + i$ and the complex number $z_{2}$ be outside the circle $C$ such that $∣ z_{1} - z_{0} ∣∣ z_{2} - z_{0} ∣ = 1$ . If $z_{0}, z_{1}$ and $z_{2}$ are collinear, then the smaller value of $∣ z_{2} ∣^{2}$ is equal to :
1. A. $\frac{3}{2}$
2. B. $\frac{5}{2}$
3. C. $\frac{13}{2}$
4. D. $\frac{7}{2}$
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Answer: B
Q232023 · 12 Apr 2023 · Shift 1· Numerical answer
Two circles in the first quadrant of radii $r_{1}$ and $r_{2}$ touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line $x + y = 2$ . Then $r_{1}^{2} + r_{2}^{2} - r_{1} r_{2}$ is equal to _________.
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Answer: 7
Q112023 · 11 Apr 2023 · Shift 1
Consider ellipses $E_{k} : k x^{2} + k^{2} y^{2} = 1, k = 1, 2, \dots, 20$ . Let $C_{k}$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse $E_{k}$ . If $r_{k}$ is the radius of the circle $C_{k}$ , then the value of $\sum_{k = 1}^{20} \frac{1}{r _{k}^{2}}$ is :
1. A.2870
2. B.3210
3. C.3320
4. D.3080
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Answer: D
Q152023 · 11 Apr 2023 · Shift 2
If the radius of the largest circle with centre $(2, 0)$ inscribed in the ellipse $x^{2} + 4 y^{2} = 36$ is $r$ , then $12 r^{2}$ is equal to :
1. A. $72$
2. B. $92$
3. C. $115$
4. D. $69$
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Answer: B
Q72023 · 10 Apr 2023 · Shift 1
An arc PQ of a circle subtends a right angle at its centre O. The mid point of the arc PQ is R. If $O P = u$ , $O R = v$ , and $O Q = α u + β v$ , then $α, β^{2}$ are the roots of the equation :
1. A. $x^{2} + x - 2 = 0$
2. B. $3 x^{2} + 2 x - 1 = 0$
3. C. $3 x^{2} - 2 x - 1 = 0$
4. D. $x^{2} - x - 2 = 0$
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Answer: D
Q122023 · 10 Apr 2023 · Shift 1
A line segment AB of length $λ$ moves such that the points A and B remain on the periphery of a circle of radius $λ$ . Then the locus of the point, that divides the line segment AB in the ratio $2 : 3$ , is a circle of radius :
1. A. $\frac{2}{3} λ$
2. B. $\frac{3}{5} λ$
3. C. $\frac{19}{7} λ$
4. D. $\frac{19}{5} λ$
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Answer: D
Q102023 · 10 Apr 2023 · Shift 2
If the points P and Q are respectively the circumcenter and the orthocentre of a $△ ABC$ , then $P A + P B + P C$ is equal to :
1. A. $QP$
2. B. $P Q$
3. C. $2 P Q$
4. D. $2 QP$
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Answer: B
Q192023 · 10 Apr 2023 · Shift 2
Let a circle of radius 4 be concentric to the ellipse $15 x^{2} + 19 y^{2} = 285$ . Then the common tangents are inclined to the minor axis of the ellipse at the angle :
1. A. $\frac{π}{4}$
2. B. $\frac{π}{3}$
3. C. $\frac{π}{6}$
4. D. $\frac{π}{12}$
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Answer: B
Q282023 · 8 Apr 2023 · Shift 1· Numerical answer
Consider circle $C_{1} : x^{2} + y^{2} - 4 x - 2 y = α - 5$ . Let its mirror image in the line $y = 2 x + 1$ be another circle $C_{2} : 5 x^{2} + 5 y^{2} - 10 f x - 10 g y + 36 = 0$ . Let $r$ be the radius of $C_{2}$ . Then $α + r$ is equal to ___________.
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Answer: 2
Q142023 · 8 Apr 2023 · Shift 2
Let $O$ be the origin and $OP$ and $OQ$ be the tangents to the circle $x^{2} + y^{2} - 6 x + 4 y + 8 = 0$ at the points $P$ and $Q$ on it. If the circumcircle of the triangle $OPQ$ passes through the point $(α, \frac{1}{2})$ , then a value of $α$ is :
1. A. $1$
2. B. $- \frac{1}{2}$
3. C. $\frac{5}{2}$
4. D. $\frac{3}{2}$
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Answer: C
Q262023 · 6 Apr 2023 · Shift 1· Numerical answer
A circle passing through the point $P (α, β)$ in the first quadrant touches the two coordinate axes at the points $A$ and $B$ . The point $P$ is above the line $A B$ . The point $Q$ on the line segment $A B$ is the foot of perpendicular from $P$ on $A B$ . If $P Q$ is equal to 11 units, then the value of $α β$ is ________.
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Answer: 121
Q172023 · 6 Apr 2023 · Shift 2
If the tangents at the points P and Q on the circle $x^{2} + y^{2} - 2 x + y = 5$ meet at the point $R (\frac{9}{4}, 2)$ , then the area of the triangle PQR is :
1. A. $\frac{13}{8}$
2. B. $\frac{5}{8}$
3. C. $\frac{5}{4}$
4. D. $\frac{13}{4}$
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Answer: B
Q252023 · 6 Apr 2023 · Shift 2· Numerical answer
For $α, β, z \in C$ and $λ > 1$ , if $λ - 1$ is the radius of the circle $∣ z - α ∣^{2} + ∣ z - β ∣^{2} = 2 λ$ , then $∣ α - β ∣$ is equal to ________.
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Answer: 2
Q72023 · 1 Feb 2023 · Shift 1
If the center and radius of the circle $\frac{z - 2}{z - 3} = 2$ are respectively $(α, β)$ and $γ$ , then $3 (α + β + γ)$ is equal to :
1. A. $12$
2. B. $10$
3. C. $11$
4. D. $9$
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Answer: A
Q192023 · 1 Feb 2023 · Shift 1
For a triangle $A B C$ , the value of $cos 2 A + cos 2 B + cos 2 C$ is least. If its inradius is 3 and incentre is M, then which of the following is NOT correct?
1. A. $M A \cdot M B = - 18$
2. B. $sin 2 A + sin 2 B + sin 2 C = sin A + sin B + sin C$
3. C.perimeter of $Δ A B C$ is $183$
4. D.area of $Δ A B C$ is $\frac{27 3}{2}$
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Answer: D
Q82023 · 31 Jan 2023 · Shift 1
A wire of length $20 m$ is to be cut into two pieces. A piece of length $l_{1}$ is bent to make a square of area $A_{1}$ and the other piece of length $l_{2}$ is made into a circle of area $A_{2}$ . If $2 A_{1} + 3 A_{2}$ is minimum then $(π l_{1}) : l_{2}$ is equal to :
1. A. $6 : 1$
2. B. $1 : 6$
3. C. $4 : 1$
4. D. $3 : 1$
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Answer: A
Q172023 · 31 Jan 2023 · Shift 1
Let a circle $C_{1}$ be obtained on rolling the circle $x^{2} + y^{2} - 4 x - 6 y + 11 = 0$ upwards 4 units on the tangent $T$ to it at the point $(3, 2)$ . Let $C_{2}$ be the image of $C_{1}$ in $T$ . Let $A$ and $B$ be the centers of circles $C_{1}$ and $C_{2}$ respectively, and $M$ and $N$ be respectively the feet of perpendiculars drawn from $A$ and $B$ on the $x$ -axis. Then the area of the trapezium AMNB is :
1. A. $2 (2 + 2)$
2. B. $4 (1 + 2)$
3. C. $3 + 22$
4. D. $2 (1 + 2)$
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Answer: B
Q152023 · 31 Jan 2023 · Shift 2
The set of all values of $a^{2}$ for which the line $x + y = 0$ bisects two distinct chords drawn from a point $P (\frac{1 + a}{2}, \frac{1 - a}{2})$ on the circle $2 x^{2} + 2 y^{2} - (1 + a) x - (1 - a) y = 0$ , is equal to :
1. A. $(0, 4]$
2. B. $(4, \infty)$
3. C. $(2, 12]$
4. D. $(8, \infty)$
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Answer: D
Q202023 · 30 Jan 2023 · Shift 1
Let $y = x + 2$ , $4 y = 3 x + 6$ and $3 y = 4 x + 1$ be three tangent lines to the circle $(x - h)^{2} + (y - k)^{2} = r^{2}$ . Then $h + k$ is equal to :
1. A. $6$
2. B. $5 (1 + 2)$
3. C. $5$
4. D. $52$
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Answer: C
Q62023 · 29 Jan 2023 · Shift 1
Let the tangents at the points $A (4, - 11)$ and $B (8, - 5)$ on the circle $x^{2} + y^{2} - 3 x + 10 y - 15 = 0$ , intersect at the point $C$ . Then the radius of the circle, whose centre is $C$ and the line joining $A$ and $B$ is its tangent, is equal to :
1. A. $\frac{2 13}{3}$
2. B. $\frac{3 3}{4}$
3. C. $13$
4. D. $213$
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Answer: A
Q272023 · 29 Jan 2023 · Shift 2· Numerical answer
A circle with centre (2, 3) and radius 4 intersects the line $x + y = 3$ at the points P and Q. If the tangents at P and Q intersect at the point $S (α, β)$ , then $4 α - 7 β$ is equal to ____.
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Answer: 11
Q282023 · 29 Jan 2023 · Shift 2· Numerical answer
A triangle is formed by the tangents at the point (2, 2) on the curves $y^{2} = 2 x$ and $x^{2} + y^{2} = 4 x$ , and the line $x + y + 2 = 0$ . If $r$ is the radius of its circumcircle, then $r^{2}$ is equal to ____.
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Answer: 10