Circles · JEE Main previous year questions

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2019 17 questions

Q92019 · 12 Apr 2019 · Shift 1
If the angle of intersection at a point where the two circles with radii $5 cm$ and $12 cm$ intersect is $9 0^{\circ}$ , then the length (in $cm$ ) of their common chord is :
1. A. $\frac{13}{5}$
2. B. $\frac{60}{13}$
3. C. $\frac{120}{13}$
4. D. $\frac{13}{2}$
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Answer: C
Q242019 · 10 Apr 2019 · Shift 1
If the circles $x^{2} + y^{2} + 5 K x + 2 y + K = 0$ and $2 (x^{2} + y^{2}) + 2 K x + 3 y - 1 = 0$ , $(K \in R)$ , intersect at the points P and Q, then the line $4 x + 5 y - K = 0$ passes through P and Q, for :
1. A.exactly two values of K
2. B.no value of K
3. C.exactly one value of K
4. D.infinitely many values of K
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Answer: B
Q272019 · 10 Apr 2019 · Shift 1
The line x = y touches a circle at the point (1,1). If the circle also passes through the point (1, -3), then its radius is :
1. A.3
2. B.2
3. C. $22$
4. D. $32$
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Answer: C
Q32019 · 10 Apr 2019 · Shift 2
A spherical iron ball of radius 10 cm is coated with a layer of ice of uniform thickness that melts at a rate of $50 c m^{3} /min$ . When the thickness of the ice is 5 cm, then the rate at which the thickness (in cm/min) of the ice decreases, is :
1. A. $\frac{5}{6 π}$
2. B. $\frac{1}{9 π}$
3. C. $\frac{1}{36 π}$
4. D. $\frac{1}{18 π}$
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Answer: D
Q222019 · 10 Apr 2019 · Shift 2
The locus of the centres of the circles, which touch the circle, $x^{2} + y^{2} = 1$ externally, also touch the y-axis and lie in the first quadrant, is :
1. A. $x = 1 + 2 y, y \geq 0$
2. B. $y = 1 + 2 x, x \geq 0$
3. C. $y = 1 + 4 x, x \geq 0$
4. D. $x = 1 + 4 y, y \geq 0$
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Answer: B
Q302019 · 9 Apr 2019 · Shift 1
If a tangent to the circle $x^{2} + y^{2} = 1$ intersects the coordinate axes at distinct points P and Q, then the locus of the mid-point of PQ is :
1. A. $x^{2} + y^{2} - 4 x^{2} y^{2} = 0$
2. B. $x^{2} + y^{2} - 2 x y = 0$
3. C. $x^{2} + y^{2} - 2 x^{2} y^{2} = 0$
4. D. $x^{2} + y^{2} - 16 x^{2} y^{2} = 0$
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Answer: A
Q212019 · 9 Apr 2019 · Shift 2
A rectangle is inscribed in a circle with a diameter lying along the line 3y = x + 7. If the two adjacent vertices of the rectangle are (–8, 5) and (6, 5), then the area of the rectangle (in sq. units) is :
1. A.72
2. B.84
3. C.56
4. D.98
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Answer: B
Q242019 · 9 Apr 2019 · Shift 2
The common tangent to the circles $x^{2} + y^{2} = 4$ and $x^{2} + y^{2} + 6 x + 8 y - 24 = 0$ also passes through the point :
1. A. $(6, - 2)$
2. B. $(4, - 2)$
3. C. $(- 4, 6)$
4. D. $(- 6, 4)$
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Answer: A
Q272019 · 9 Apr 2019 · Shift 2
The area (in sq. units) of the smaller of the two circles that touch the parabola, $y^{2} = 4 x$ at the point (1, 2) and the x-axis is :-
1. A. $4 π (3 + 2)$
2. B. $8 π (2 - 2)$
3. C. $8 π (3 - 22)$
4. D. $4 π (2 - 2)$
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Answer: C
Q252019 · 8 Apr 2019 · Shift 2
Given that the slope of the tangent to a curve y = y(x) at any point (x, y) is $\frac{2 y}{x ^{2}}$ . If the curve passes through the centre of the circle $x^{2} + y^{2} - 2 x - 2 y = 0$ , then its equation is :
1. A. $x lo g_{e} ∣ y ∣ = 2 (x - 1)$
2. B. $x^{2} lo g_{e} ∣ y ∣ = - 2 (x - 1)$
3. C. $x lo g_{e} ∣ y ∣ = x - 1$
4. D. $x lo g_{e} ∣ y ∣ = - 2 (x - 1)$
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Answer: A
Q292019 · 8 Apr 2019 · Shift 2
The tangent and the normal lines at the point $(3, 1)$ to the circle $x^{2} + y^{2} = 4$ and the x-axis form a triangle. The area of this triangle (in square units) is :
1. A. $\frac{4}{3}$
2. B. $\frac{1}{3}$
3. C. $\frac{2}{3}$
4. D. $\frac{1}{3}$
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Answer: C
Q262019 · 12 Jan 2019 · Shift 1
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 :
1. A. $(23, 31)$
2. B. $(2, 17)$
3. C. $[13, 23]$
4. D. $[12, 21]$
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Answer: D
Q292019 · 12 Jan 2019 · Shift 1
Let $C_{1}$ and $C_{2}$ be the centres of the circles $x^{2} + y^{2} - 2 x - 2 y - 2 = 0$ and $x^{2} + y^{2} - 6 x - 6 y + 14 = 0$ respectively. If P and Q are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral $P C_{1} Q C_{2}$ is :
1. A.4
2. B.6
3. C.9
4. D.8
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Answer: A
Q72019 · 12 Jan 2019 · Shift 2
If a circle of radius R passes through the origin O and intersects the coordinate axes at A and B, then the locus of the foot of perpendicular from O on AB is :
1. A. $(x^{2} + y^{2})^{2} = 4 R^{2} x^{2} y^{2}$
2. B. $(x^{2} + y^{2}) (x + y) = R^{2} x y$
3. C. $(x^{2} + y^{2})^{2} = 4 R x^{2} y^{2}$
4. D. $(x^{2} + y^{2})^{3} = 4 R^{2} x^{2} y^{2}$
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Answer: D
Q202019 · 11 Jan 2019 · Shift 2
A circle cuts a chord of length 4a on the x-axis and passes through a point on the y-axis, distant 2b from the origin. Then the locus of the centre of this circle, is :
1. A.an ellipse
2. B.a parabola
3. C.a hyperbola
4. D.a straight line
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Answer: B
Q272019 · 9 Jan 2019 · Shift 1
Three circles of radii $a, b, c$ $(a < b < c)$ touch each other externally. If they have x-axis as a common tangent, then :
1. A.a, b, c are in A.P.
2. B. $a, b, c$ are in A.P
3. C. $\frac{1}{b} + \frac{1}{c} = \frac{1}{a}$
4. D. $\frac{1}{b} = \frac{1}{a} + \frac{1}{c}$
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Answer: C
Q302019 · 9 Jan 2019 · Shift 1
Equation of a common tangent to the circle, $x^{2} + y^{2} - 6 x = 0$ and the parabola, $y^{2} = 4 x$ is :
1. A. $23 y = 12 x + 1$
2. B. $3 y = x + 3$
3. C. $23 y = - x - 12$
4. D. $3 y = 3 x + 1$
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Answer: B