Circles · JEE Main previous year questions

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2022 26 questions

Q252022 · 29 Jul 2022 · Shift 2· Numerical answer
Let $A B$ be a chord of length 12 of the circle $(x - 2)^{2} + (y + 1)^{2} = \frac{169}{4}$ . If tangents drawn to the circle at points $A$ and $B$ intersect at the point $P$ , then five times the distance of point $P$ from chord $A B$ is equal to ________.
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Answer: 72
Q62022 · 28 Jul 2022 · Shift 1
For $t \in (0, 2 π)$ , if $ABC$ is an equilateral triangle with vertices $A (sin t, - cos t)$ , $B (cos t, sin t)$ and $C (a, b)$ such that its orthocentre lies on a circle with centre $(1, \frac{1}{3})$ , then $(a^{2} - b^{2})$ is equal to :
1. A. $\frac{8}{3}$
2. B.8
3. C. $\frac{77}{9}$
4. D. $\frac{80}{9}$
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Answer: B
Q102022 · 28 Jul 2022 · Shift 1
Let $C$ be the centre of the circle $x^{2} + y^{2} - x + 2 y = \frac{11}{4}$ and $P$ be a point on the circle. A line passes through the point $C$ , makes an angle of $\frac{π}{4}$ with the line $C P$ and intersects the circle at the points $Q$ and $R$ . Then the area of the triangle $P QR$ (in $unit^{2}$ ) is :
1. A.2
2. B. $22$
3. C. $8 sin (\frac{π}{8})$
4. D. $8 cos (\frac{π}{8})$
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Answer: B
Q162022 · 28 Jul 2022 · Shift 2
Let the tangents at two points A and B on the circle $x^{2} + y^{2} - 4 x + 3 = 0$ meet at origin $O (0, 0)$ . Then the area of the triangle OAB is :
1. A. $\frac{3 3}{2}$
2. B. $\frac{3 3}{4}$
3. C. $\frac{3}{2 3}$
4. D. $\frac{3}{4 3}$
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Answer: B
Q282022 · 28 Jul 2022 · Shift 2· Numerical answer
Two tangent lines $l_{1}$ and $l_{2}$ are drawn from the point $(2, 0)$ to the parabola $2 y^{2} = - x$ . If the lines $l_{1}$ and $l_{2}$ are also tangent to the circle $(x - 5)^{2} + y^{2} = r$ , then $17 r$ is equal to _________.
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Answer: 9
Q142022 · 27 Jul 2022 · Shift 1
If the circle $x^{2} + y^{2} - 2 g x + 6 y - 19 c = 0, g, c \in R$ passes through the point $(6, 1)$ and its centre lies on the line $x - 2 cy = 8$ , then the length of intercept made by the circle on $x$ -axis is :
1. A. $11$
2. B.4
3. C.3
4. D. $223$
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Answer: D
Q162022 · 27 Jul 2022 · Shift 1
Let $P (a, b)$ be a point on the parabola $y^{2} = 8 x$ such that the tangent at $P$ passes through the centre of the circle $x^{2} + y^{2} - 10 x - 14 y + 65 = 0$ . Let $A$ be the product of all possible values of $a$ and $B$ be the product of all possible values of $b$ . Then the value of $A + B$ is equal to :
1. A.0
2. B.25
3. C.40
4. D.65
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Answer: D
Q142022 · 27 Jul 2022 · Shift 2
A circle $C_{1}$ passes through the origin $O$ and has diameter $4$ on the positive $x$ -axis. The line $y = 2 x$ gives a chord $OA$ of circle $C_{1}$ . Let $C_{2}$ be the circle with $OA$ as a diameter. If the tangent to $C_{2}$ at the point $A$ meets the $x$ -axis at $P$ and $y$ -axis at $Q$ , then $QA : AP$ is equal to :
1. A. $1 : 4$
2. B. $1 : 5$
3. C. $2 : 5$
4. D. $1 : 3$
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Answer: A
Q302022 · 25 Jul 2022 · Shift 2· Numerical answer
If the circles $x^{2} + y^{2} + 6 x + 8 y + 16 = 0$ and $x^{2} + y^{2} + 2 (3 - 3) x + 2 (4 - 6) y = k + 63 + 86$ , $k > 0$ , touch internally at the point $P (α, β)$ , then $(α + 3)^{2} + (β + 6)^{2}$ is equal to ___________.
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Answer: 25
Q152022 · 30 Jun 2022 · Shift 1
Let $\frac{d y}{d x} = \frac{a x - b y + a}{b x + cy + a}$ , $a, b, c \in R$ , represents a circle with center $(α, β)$ . Then, $α + 2 β$ is equal to :
1. A. $- 1$
2. B.0
3. C.1
4. D.2
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Answer: C
Q162022 · 30 Jun 2022 · Shift 1
Consider three circles: $C_{1} : x^{2} + y^{2} = r^{2}$ $C_{2} : (x - 1)^{2} + (y - 1)^{2} = r^{2}$ $C_{3} : (x - 2)^{2} + (y - 1)^{2} = r^{2}$ If a line $L : y = m x + c$ be a common tangent to $C_{1}$ , $C_{2}$ and $C_{3}$ such that $C_{1}$ and $C_{3}$ lie on one side of line L while $C_{2}$ lies on other side, then the value of $20 (r^{2} + c)$ is equal to :
1. A.23
2. B.15
3. C.12
4. D.6
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Answer: D
Q232022 · 28 Jun 2022 · Shift 2· Numerical answer
If one of the diameters of the circle $x^{2} + y^{2} - 22 x - 62 y + 14 = 0$ is a chord of the circle $(x - 22)^{2} + (y - 22)^{2} = r^{2}$ , then the value of $r^{2}$ is equal to ________.
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Answer: 10
Q172022 · 27 Jun 2022 · Shift 1
Let $\frac{d y}{d x} = \frac{a x - b y + a}{b x + cy + a}$ , where $a, b, c$ are constants, represent a circle passing through the point $(2, 5)$ . Then the shortest distance of the point $(11, 6)$ from this circle is :
1. A.10
2. B.8
3. C.7
4. D.5
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Answer: B
Q262022 · 27 Jun 2022 · Shift 1· Numerical answer
A rectangle R with end points of one of its sides as $(1, 2)$ and $(3, 6)$ is inscribed in a circle. If the equation of a diameter of the circle is $2 x - y + 4 = 0$ , then the area of R is ____.
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Answer: 16
Q292022 · 27 Jun 2022 · Shift 1· Numerical answer
A circle of radius 2 unit passes through the vertex and the focus of the parabola $y^{2} = 2 x$ and touches the parabola $y = (x - \frac{1}{4})^{2} + α$ , where $α > 0$ . Then $(4 α - 8)^{2}$ is equal to ____.
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Answer: 63
Q82022 · 27 Jun 2022 · Shift 2
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 :
1. A.an empty set
2. B. $(6, \frac{65}{9}]$
3. C. $[\frac{80}{9}, 10)$
4. D. $(9, \frac{92}{9}]$
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Answer: D
Q302022 · 27 Jun 2022 · Shift 2· Numerical answer
Let a circle $C$ of radius $5$ lie below the x-axis. The line $L_{1} : 4 x + 3 y + 2 = 0$ passes through the centre $P$ of the circle $C$ and intersects the line $L_{2} = 3 x - 4 y - 11 = 0$ at $Q$ . The line $L_{2}$ touches $C$ at the point $Q$ . Then the distance of $P$ from the line $5 x - 12 y + 51 = 0$ is ________.
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Answer: 11
Q152022 · 26 Jun 2022 · Shift 1
Let $C$ be a circle passing through the points $A (2, - 1)$ and $B (3, 4)$ . The line segment $A B$ is not a diameter of $C$ . If $r$ is the radius of $C$ and its centre lies on the circle $(x - 5)^{2} + (y - 1)^{2} = \frac{13}{2}$ , then $r^{2}$ is equal to :
1. A.32
2. B. $\frac{65}{2}$
3. C. $\frac{61}{2}$
4. D.30
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Answer: B
Q162022 · 25 Jun 2022 · Shift 1
Let a circle $C$ touch the lines $L_{1} : 4 x - 3 y + K_{1} = 0$ and $L_{2} : 4 x - 3 y + K_{2} = 0$ , $K_{1}, K_{2} \in R$ . If a line passing through the centre of the circle $C$ intersects $L_{1}$ at $(- 1, 2)$ and $L_{2}$ at $(3, - 6)$ , then the equation of the circle $C$ is :
1. A. $(x - 1)^{2} + (y - 2)^{2} = 4$
2. B. $(x + 1)^{2} + (y - 2)^{2} = 4$
3. C. $(x - 1)^{2} + (y + 2)^{2} = 16$
4. D. $(x - 1)^{2} + (y - 2)^{2} = 16$
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Answer: C
Q172022 · 25 Jun 2022 · Shift 1
Let $a$ , $b$ and $c$ be the length of sides of a triangle $ABC$ such that $\frac{a + b}{7} = \frac{b + c}{8} = \frac{c + a}{9}$ . If $r$ and $R$ are the radius of incircle and radius of circumcircle of the triangle $ABC$ , respectively, then the value of $\frac{R}{r}$ is equal to :
1. A. $\frac{5}{2}$
2. B. $2$
3. C. $\frac{3}{2}$
4. D. $1$
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Answer: A
Q192022 · 25 Jun 2022 · Shift 1
If $y = m_{1} x + c_{1}$ and $y = m_{2} x + c_{2}$ , $m_{1} \neq = m_{2}$ are two common tangents of circle $x^{2} + y^{2} = 2$ and parabola $y^{2} = x$ , then the value of $8∣ m_{1} m_{2} ∣$ is equal to :
1. A. $3 + 42$
2. B. $- 5 + 62$
3. C. $- 4 + 32$
4. D. $7 + 62$
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Answer: C
Q232022 · 25 Jun 2022 · Shift 1· Numerical answer
Let the abscissae of the two points $P$ and $Q$ be the roots of $2 x^{2} - r x + p = 0$ and the ordinates of $P$ and $Q$ be the roots of $x^{2} - s x - q = 0$ . If the equation of the circle described on $PQ$ as diameter is $2 (x^{2} + y^{2}) - 11 x - 14 y - 22 = 0$ , then $2 r + s - 2 q + p$ is equal to .
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Answer: 7
Q142022 · 25 Jun 2022 · Shift 2
The line $y = x + 1$ meets the ellipse $\frac{x ^{2}}{4} + \frac{y ^{2}}{2} = 1$ at two points $P$ and $Q$ . If $r$ is the radius of the circle with $PQ$ as diameter then $(3 r)^{2}$ is equal to :
1. A. $20$
2. B. $12$
3. C. $11$
4. D. $8$
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Answer: A
Q172022 · 25 Jun 2022 · Shift 2
A circle touches both the y-axis and the line $x + y = 0$ . Then the locus of its center is :
1. A. $y = 2 x$
2. B. $x = 2 y$
3. C. $y^{2} - x^{2} = 2 x y$
4. D. $x^{2} - y^{2} = 2 x y$
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Answer: D
Q272022 · 24 Jun 2022 · Shift 2· Numerical answer
Let a circle $C : (x - h)^{2} + (y - k)^{2} = r^{2}$ , $k > 0$ , touch the x-axis at $(1, 0)$ . If the line $x + y = 0$ intersects the circle C at P and Q such that the length of the chord PQ is 2, then the value of $h + k + r$ is equal to ____.
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Answer: 7