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JEE • MATHEMATICS • CIRCLES

Circles practice · Easy

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241 Easy
241 of 480
Application +4 | -1
The line 3x−2y=k3x - 2y = k meets the circle x2+y2=4r2{x^2} + {y^2} = 4{r^2} at only one point, if k2{k^2} is
View answer and explanation
Correct option: B
52 r252\,{r^2}
Option B (52 r252\,{r^2}) is correct because it is the value of k2{k^2} when the line is tangent to the circle.
242 Easy
242 of 480
Application +4 | -1
The number of real tangents that can be drawn from (2,2)(2,2) to the circle x2+y2−6x−4y+3=0x^2+y^2-6 x-4 y+3=0, is
View answer and explanation
Correct option: A
0
Option A (0) is correct because the point (2,2) lies inside the circle, and thus no real tangents can be drawn.
243 Easy
243 of 480
Application +4 | -1
If m1{m_1} and m2{m_2} are the slopes of tangents to the circle x2+y2=4  {x^2} + {y^2} = 4\; from the point (3,2)\left( {3,2} \right), then m1−m2  {m_1} - {m_2}\; is equal to
View answer and explanation
Correct option: B
125\frac{{12}}{5}
Option B (12/5) is correct because it is the difference between the slopes of the tangents calculated using the equation of the pair of tangents from a point to a circle.
244 Easy
244 of 480
Application +4 | -1
The point on the line x=4x=4 from which the tangents drawn to the circle x2+y2=18x^2+y^2=18 are at right angle is
View answer and explanation
Correct option: D
(4,25)(4,2 \sqrt{5})
Option A (3) is correct because by equating the derivatives and using the given point, we can solve for a and b and then find the value of (a/2 - b).
245 Easy
245 of 480
Application +4 | -1
The number of circles that touch all the straight lines x+y−4=0,x−y+2=0x + y - 4 = 0,x - y + 2 = 0 and y=2,y = 2, is
View answer and explanation
Correct option: D
4
Option D (4) is correct because there are four circles that can touch all three lines: one incircle and three excircles of the triangle formed by the lines.
246 Easy
246 of 480
Application +4 | -1
Find the equation of tangents to the circle x2+y2−6x+4y−x^2+y^2-6 x+4 y- 12=012=0 which are parallel to the line 4x+3y+5=04 x+3 y+5=0.
View answer and explanation
Correct option: B
4x+3y+19=0&4x+3y−31=04 x+3 y+19=0 \& 4 x+3 y-31=0
Option B (4x+3y+19=0&4x+3y−31=04 x+3 y+19=0 \& 4 x+3 y-31=0) is correct because these are the equations of the tangents to the given circle that are parallel to the given line, found by using the distance from the center to the tangent equals the radius.
247 Easy
247 of 480
Application +4 | -1
The radius of the larger circle lying in the first quadrant and touching the line 4x+3y−12=04x + 3y - 12 = 0 and the coordinate axes, is
View answer and explanation
Correct option: B
6
The correct answer is B (6) because it represents the radius of the larger circle that satisfies the given conditions.
248 Easy
248 of 480
Analyzing +4 | -1
The area (in sq units) of the smaller of the two circles that touch the parabola, y2=4x{y^2} = 4x at the point (1, 2) and the X-axis is
View answer and explanation
Correct option: A
8π(3−22)8\pi (3 - 2\sqrt 2 )
Option A (8π(3−22)8\pi (3 - 2\sqrt 2 )) is correct because it represents the area of the smaller circle that touches the parabola at (1, 2) and the x-axis, calculated using geometric properties and calculus.
249 Easy
249 of 480
Application +4 | -1
Two perpendicular tangents to the circle x2+y2=a2{x^2} + {y^2} = {a^2} meet at PP. Then, the locus of PP has the equation
View answer and explanation
Correct option: A
x2+y2=2a2{x^2} + {y^2} = 2{a^2}
Option A (x2+y2=2a2{x^2} + {y^2} = 2{a^2}) is correct because the locus of the point of intersection of two perpendicular tangents to a circle is the director circle, which has the equation x2+y2=2a2{x^2} + {y^2} = 2{a^2}.
250 Easy
250 of 480
Application +4 | -1
The locus of the point P(h,k)P(h, k) for which the line hx+ky=1h x+k y=1 touches the circle x2+y2=4x^2+y^2=4, is
View answer and explanation
Correct option: A
a circle
Option A (a circle) is correct because the condition for tangency leads to the equation h2+k2=14h^2 + k^2 = \frac{1}{4}, which represents a circle.
251 Easy
251 of 480
Analyzing +4 | -1
If two circles a(x2+y2)+bx+cy=0a\left(x^2+y^2\right)+b x+c y=0 and A(x2+y2)+Bx+Cy=0A\left(x^2+y^2\right)+B x+C y=0 touch each other then
View answer and explanation
Correct option: B
bC=cBb C=c B
Option B (bC=cBb C=c B) is correct because the circles touch each other if they have a common tangent at the origin, leading to the given condition.
252 Easy
252 of 480
Application +4 | -1
The locus of a point which moves such that the tangents from it to the two circles x2+y2−5x−3=0x^2+y^2-5 x-3=0 and 3x2+3y2+2x+4y−6=03 x^2+3 y^2+2 x+4 y-6=0 are equal, is given by
View answer and explanation
Correct option: B
17x+4y+3=017 x+4 y+3=0
Option B (17x+4y+3=017 x+4 y+3=0) is correct because it represents the radical axis of the two circles, which is the locus of points with equal tangent lengths.
253 Easy
253 of 480
Analyzing +4 | -1
If the circles x2+y2+2ax+b=0x^2+y^2+2 a x+b=0 and x2+y2+2cx+b=0x^2+y^2+2 c x+b=0 touch each other, then
View answer and explanation
Correct option: C
b=0b=0
Option C (b=0b=0) is correct because the condition for the circles to touch each other requires the common tangent to have equal roots, leading to b=0.
254 Easy
254 of 480
Analyzing +4 | -1
The set of values of cc, so that the equations y=∣x∣+cy=|x|+c and x2+y2−8∣x∣−9=0x^2+y^2-8|x|-9=0 have no solution, is
View answer and explanation
Correct option: D
(52−4,∞)(5 \sqrt{2}-4, \infty)
Option D ((52−4,∞)(5 \sqrt{2}-4, \infty)) is correct because the condition for no solution is that the constant c in y=|x|+c must be greater than the y-intercept of the tangent to the circle parallel to y=x.
255 Easy
255 of 480
Understanding +4 | -1
The circle (x−r)2+(y−r)2=r2(x-r)^2+(y-r)^2=r^2 touches
View answer and explanation
Correct option: C
both the coordinate axes
Option C (both the coordinate axes) is correct because the distance of the center from both axes is equal to the radius.

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256 Easy
256 of 480
Application +4 | -1
Equation of a circle with centre (4,3)(4,3) touching the circle x2+y2=1x^2+y^2=1 is
View answer and explanation
Correct option: D
x2+y2−8x−6y+9=0x^2+y^2-8 x-6 y+9=0
Option D (x2+y2−8x−6y+9=0x^2+y^2-8 x-6 y+9=0) is the correct answer because it represents a circle with center (4,3) that touches the circle x2+y2=1x^2 + y^2 = 1 externally.
257 Easy
257 of 480
Application +4 | -1
The locus of the point of intersection of the tangents to the circle x=rcos⁡θ,y=rsin⁡θx=r \cos \theta, y=r \sin \theta at points whose parametric angles differ by π3\frac{\pi}{3} is
View answer and explanation
Correct option: D
3(x2+y2)=4r23\left(x^2+y^2\right)=4 r^2
Option D (3(x2+y2)=4r23(x^2+y^2)=4 r^2) is correct because it represents the locus of the intersection point of tangents to the circle whose parametric angles differ by π3\frac{\pi}{3}.
258 Easy
258 of 480
Application +4 | -1
If (a,b)(a, b) is a point on the chord ABA B of the circle, where the ends of the chord are A=(2,−3)A=(2,-3) and B=(3,2)B=(3,2), then
View answer and explanation
Correct option: B
a∈(2,3),b∈(−3,2)a \in(2,3), b \in(-3,2)
Option B (a∈(2,3),b∈(−3,2)a \in(2,3), b \in(-3,2)) is correct because if (a, b) lies on the line segment AB, then 'a' must lie between the x-coordinates of A and B, and 'b' must lie between the y-coordinates of A and B.
259 Easy
259 of 480
Analyzing +4 | -1
If a circle having centre at (α,β)(\alpha, \beta) radius rr completely lies with in two lines x+y=2x+y=2 and x+y=−2x+y=-2, then, min⁡.(∣α+β+2∣,∣α+β−2∣)\min .(|\alpha+\beta+2|,|\alpha+\beta-2|) is
View answer and explanation
Correct option: A
greater than 2r\sqrt{2} r
Option A (greater than 2r\sqrt{2} r) is correct because the minimum distance of the center from either line must be greater than the radius of the circle.
260 Easy
260 of 480
Application +4 | -1
The value of cc for which line y=2x+cy=2 x+c is tangent to circle x2+y2=(5)2x^2+y^2=(\sqrt{5})^2 is
View answer and explanation
Correct option: B
±5\pm 5
Option B is correct because the value of c for the line y = 2x + c to be tangent to the circle x2 + y2 = (sqrt(5))2 is ±5, which is found using the tangency condition.
261 Easy
261 of 480
Application +4 | -1
The centre of a circle is (2,−3)\left( {2, - 3} \right) and the circumference is 10π10\pi. Then, the equation of the circle is
View answer and explanation
Correct option: C
x2+y2−4x+6y−12=0{x^2} + {y^2} - 4x + 6y - 12 = 0
Option C (x2+y2−4x+6y−12=0{x^2} + {y^2} - 4x + 6y - 12 = 0) is correct because it represents the equation of a circle with center (2, -3) and radius 5, derived from the given circumference.
262 Easy
262 of 480
Application +4 | -1
The equation of the circle passing through (4, 5) and having the centre (2, 2), is
View answer and explanation
Correct option: B
x2+y2−4x−4y−5=0{x^2} + {y^2} - 4x - 4y - 5 = 0
Option B (x2+y2−4x−4y−5=0{x^2} + {y^2} - 4x - 4y - 5 = 0) is correct because it represents the equation of the circle passing through (4, 5) and having the center (2, 2).
263 Easy
263 of 480
Application +4 | -1
Extremities of a diagonal of a rectangle are (0,0)\left( {0,0} \right) and (4,3)\left( {4,3} \right). The equations of the tangents to the circumcircle of the rectangle which are parallel to the diagonal are
View answer and explanation
Correct option: B
6x−8y±25=06x - 8y \pm 25 = 0
Option B (6x−8y±25=06x - 8y \pm 25 = 0) is correct because it represents the tangent lines to the circumcircle that are parallel to the given diagonal.
264 Easy
264 of 480
Application +4 | -1
If P1,P2,P3{P_1},{P_2},{P_3} are the perimeter of the three circles x2+y2+8x−6y=0,  4x2+4y2−4x−12y−186=0{x^2} + {y^2} + 8x - 6y = 0,\;4{x^2} + 4{y^2} - 4x - 12y - 186 = 0 and x2+y2−6x+6y−9=0{x^2} + {y^2} - 6x + 6y - 9 = 0 respectively, then
View answer and explanation
Correct option: B
P1<P3<P2{P_1} < {P_3} < {P_2}
Option B (P1<P3<P2{P_1} < {P_3} < {P_2}) is correct because it accurately compares the perimeters of the three circles based on their calculated radii.
265 Easy
265 of 480
Application +4 | -1
If lx+my=1l x+m y=1 touches the circle x2+y2=a2x^2+y^2=a^2 then the point (l,m)(l, m) lies on the circle
View answer and explanation
Correct option: A
x2+y2=a−2x^2+y^2=a^{-2}
Option A (x2+y2=a−2x^2+y^2=a^{-2}) is correct because the condition for the line to touch the circle leads to the point (l, m) lying on this circle.
266 Easy
266 of 480
Application +4 | -1
The coordinates of the middle point of the chord intercepted on line lx+my+n=0l x+m y+n=0 by the circle x2+y2=a2x^2+y^2=a^2 are
View answer and explanation
Correct option: B
(−ln⁡l2+m2,−mnl2+m2)\left(\frac{-\ln }{l^2+m^2}, \frac{-m n}{l^2+m^2}\right)
Option B ((−ln⁡l2+m2,−mnl2+m2)\left(\frac{-\ln }{l^2+m^2}, \frac{-m n}{l^2+m^2}\right)) is correct because it represents the midpoint of the chord intercepted by the line on the circle.
267 Easy
267 of 480
Application +4 | -1
Equation of a circle S(x,y)=0,(S(2,3)=16)S(x, y)=0,(S(2,3)=16) which touches the line 3x+4y−7=03 x+4 y-7=0 at (1,1)(1,1) is given by
View answer and explanation
Correct option: A
x2+y2+x+2y−5=0x^2+y^2+x+2 y-5=0
Option A (x2+y2+x+2y−5=0x^2+y^2+x+2 y-5=0) is correct because it satisfies the given conditions of touching the line at (1,1) and S(2,3)=16.
268 Easy
268 of 480
Application +4 | -1
The equation of the circle touching x=0,y=0x = 0,y = 0 and x=4x = 4 is
View answer and explanation
Correct option: D
x2+y2−4x−4y+4=0{x^2} + {y^2} - 4x - 4y + 4 = 0
Option D (x2+y2−4x−4y+4=0{x^2} + {y^2} - 4x - 4y + 4 = 0) is correct because it represents a circle touching x = 0, y = 0, and x = 4, which means its center is at (2, 2) and its radius is 2.
269 Easy
269 of 480
Application +4 | -1
Find the length of the tangent drawn from the point (5,1)(5,1) to the circle x2+y2+6x−4y−3=0x^2+y^2+6 x-4 y-3=0.
View answer and explanation
Correct option: C
7
Option C (7) is correct because the length of the tangent from a point to a circle is calculated by substituting the point's coordinates into the circle's equation and taking the square root.
270 Easy
270 of 480
Application +4 | -1
Combined equation of tangents drawn to x2+y2=1x^2+y^2=1 from (−1,−1)(-1,-1), is
View answer and explanation
Correct option: D
x+y+xy+1=0x+y+x y+1=0
Option D (x+y+xy+1=0x+y+x y+1=0) is correct because the combined equation of the tangents is found by multiplying the equations of the individual tangents.
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