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.
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 ›
Q2.
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 ›
Q3.
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 ›