Circle Questions with Solutions

19 questions7 easy9 medium3 hard

In coordinate geometry, a circle is the set of points at a fixed distance (the radius) from a fixed point (the centre). This topic covers the standard equation (x - h)^2 + (y - k)^2 = r^2, the general equation x^2 + y^2 + 2gx + 2fy + c = 0 with centre (-g, -f), the diameter form, and parametric equations. Exam questions also test the position of a point relative to a circle, the condition for a line to be a tangent, equations of tangents and normals, the length of a tangent from an external point, and the intersection of two circles. The fastest approach is often to compare the distance from the centre to a line with the radius: equal means tangent, smaller means the line cuts the circle. Practise the circle questions below, each with a step-by-step solution.

All Questions Easy Medium Hard

Q1.
Segments of the lines $2 x + 3 y = 1$ and $4 x - 3 y = 11$ are diameters of a circle of area 153.94 square units. Then the equation of this circle with integer radius is:
A. $x^{2} + y^{2} + 4 x - 2 y - 44 = 0$ B. $x^{2} + y^{2} + 4 x - 2 y + 44 = 0$ C. $x^{2} + y^{2} - 4 x + 2 y - 44 = 0$ D. $x^{2} + y^{2} - 4 x + 2 y + 44 = 0$
easyAsked in NIMCET 2026View solution ›
Q2.
The equation $x^{2} + y^{2} + 4 x + 6 y + 13 = 0$ represents
A. a pair of coincident linesB. a pair of concurrent straight linesC. a parabolaD. a point circle
easyAsked in AMU MCA 2016View solution ›
Q3.
Let AB be a chord of a circle $x^{2} + y^{2} = r^{2}$ subtending a right angle at the centre. Then, the locus of the centroid of the triangle PAB as P moves on the circle is
A. ParabolaB. A circleC. An ellipseD. A pair of straight line
mediumAsked in MAH MCA CET 2024View solution ›
Q4.
The lines $3 x - 4 y + 4 = 0$ and $6 x - 8 y - 7 = 0$ are tangents to the same circle. The radius of this circle is
A. 3/2B. 3/4C. 4/5D. 7/10
mediumAsked in NIMCET 2013View solution ›
Q5.
Find the equation of the circle which passes through (-1, 1) and (2, 1), and has its centre on the line $x + 2 y + 3 = 0$ .
A. $2 x^{2} + 2 y^{2} - 2 x + 7 y - 13 = 0$ B. $x^{2} + y^{2} - 2 x + 7 y - 13 = 0$ C. $2 x^{2} + 2 y^{2} + 2 x + 7 y - 13 = 0$ D. $x^{2} + y^{2} + 2 x + 7 y - 13 = 0$
mediumAsked in NIMCET 2013View solution ›
Q6.
If the circles $x^{2} + y^{2} + 2 x + 2 k y + 6 = 0$ and $x^{2} + y^{2} + 2 k y + k = 0$ intersect orthogonally, then $k$ is
A. $2$ or $- \frac{3}{2}$ B. $- 2$ or $- \frac{3}{2}$ C. $2$ or $\frac{3}{2}$ D. $- 2$ or $\frac{3}{2}$
mediumAsked in NIMCET 2012View solution ›
Q7.
The lines $2 x + 3 y - 6 = 0$ and $9 x + 6 y - 18 = 0$ cut the coordinate axes in concyclic points. The centre of the circle is
A. $(1, 2)$ B. $(2, 3)$ C. $(5, 5)$ D. $(\frac{5}{2}, \frac{5}{2})$
mediumAsked in NIMCET 2011View solution ›
Q8.
As $a$ varies over all real numbers, the possible numbers of distinct real solutions $(x, y)$ of the system $x^{2} = y^{2}$ , $(x - a)^{2} + y^{2} = 1$ are
A. $0, 1, 2, 3, 4$ B. $0, 1, 3$ C. $0, 1, 2$ D. $0, 2, 3, 4$
hardAsked in NIMCET 2011View solution ›
Q9.
The circle $x^{2} + y^{2} = 9$ is contained in the circle $x^{2} + y^{2} - 6 x - 8 y + 25 = c^{2}$ if
A. $c = 2$ B. $c = 3$ C. $c = 5$ D. $c = 10$
mediumAsked in NIMCET 2010View solution ›
Q10.
The number of common tangents to the circles $x^{2} + y^{2} = 4$ and $x^{2} + y^{2} - 6 x - 8 y = 24$ is
A. 0B. 1C. 3D. 4
mediumAsked in NIMCET 2021View solution ›
Q11.
A circle touches the X-axis and also touches another circle with centre at (0, 3) and radius 2. Then the locus of the centre of the first circle is
A. a parabolaB. a hyperbolaC. a circleD. an ellipse
mediumAsked in NIMCET 2015View solution ›
Q12.
The radius of the circle passing through the foci of the ellipse $\frac{x ^{2}}{16} + \frac{y ^{2}}{9} = 1$ and having its centre at (0, 3) is
A. 4 unitsB. 3 unitsC. $12$ unitsD. $\frac{7}{2}$ units
easyAsked in NIMCET 2015View solution ›
Q13.
If two circles $x^{2} + y^{2} + 2 g x + 2 f y = 0$ and $x^{2} + y^{2} + 2 g' x + 2 f' y = 0$ touch each other, then which of the following is true?
A. $g f = g' f'$ B. $g' f = g f'$ C. $g g' = f f'$ D. None of these
mediumAsked in NIMCET 2015View solution ›
Q14.
The point of intersection of the circle $x^{2} + y^{2} + 10 x - 12 y + 51 = 0$ and the line $x + 3 y = 3$ is
A. $(- 6, 3)$ B. $(3, - 6)$ C. $(6, - 3)$ D. $(- 3, 6)$
easyAsked in NIMCET 2018View solution ›
Q15.
The circles whose equations are $x^{2} + y^{2} + c^{2} = 2 a x$ and $x^{2} + y^{2} + c^{2} - 2 b y = 0$ will touch one another externally if
A. $\frac{1}{b ^{2}} + \frac{1}{c ^{2}} = \frac{1}{a ^{2}}$ B. $\frac{1}{c ^{2}} + \frac{1}{a ^{2}} = \frac{1}{b ^{2}}$ C. $\frac{1}{a ^{2}} + \frac{1}{b ^{2}} = \frac{1}{c ^{2}}$ D. None of these
hardAsked in NIMCET 2018View solution ›
Q16.
The equation of the circle passing through the point (4, 6) and whose diameters are along $x + 2 y - 5 = 0$ and $3 x - y - 1 = 0$ is
A. $x^{2} + y^{2} - 2 x - 6 y - 20 = 0$ B. $x^{2} + y^{2} - 6 x - 2 y - 20 = 0$ C. $x^{2} + y^{2} - 2 x - 4 y - 20 = 0$ D. $x^{2} + y^{2} - 4 x - 2 y - 20 = 0$
easyAsked in NIMCET 2019View solution ›
Q17.
For the two circles $x^{2} + y^{2} = 16$ and $x^{2} + y^{2} - 2 y = 0$ , there is/are
A. One pair of common tangentsB. Two pairs of common tangentsC. Three pairs of common tangentsD. No common tangents
easyAsked in NIMCET 2019View solution ›
Q18.
A circle with its center in the first quadrant touches both the coordinate axes and the line $x - y - 2 = 0$ . Then the area of the circle is
A. $\frac{π}{2}$ B. $4 π$ C. $π$ D. $2 π$
easyAsked in NIMCET 2025View solution ›
Q19.
The circle $x^{2} + y^{2} + α x + β y + γ = 0$ is the image of the circle $x^{2} + y^{2} - 6 x - 10 y + 30 = 0$ across the line $3 x + y = 2$ . The value of $[α + β + γ]$ is (where $[\cdot]$ represents the floor function)
A. 20B. 22C. 23D. 21
hardAsked in NIMCET 2025View solution ›