It is always possible to draw a circle passing through ______ non-collinear points in a plane.
- (a)3
- (b)6
- (c)5
- (d)4
Answer
Why
Correct — A. Through three points that do not lie on one straight line, exactly one circle can always be drawn.
Join two pairs of the points and draw the perpendicular bisector of each segment.
A perpendicular bisector holds every point equidistant from that segment's two ends.
The two bisectors meet at one point, the circumcentre, equidistant from all three points.
That common distance is the radius, so the circle exists and is unique → option (a)
Had the three points been collinear the two bisectors would be parallel and never meet, which is exactly why the stem says non-collinear.
Why the others are wrong
- (b)6 — Six points lie on one circle only when they happen to be concyclic. Choose six non-collinear points freely and no circle passes through all of them, so the word always fails.
- (c)5 — A circle carries only three degrees of freedom — two for the centre, one for the radius. Five points impose five conditions on three unknowns, so in general no such circle exists.
- (d)4 — Four is the first case that breaks. Any three of them already fix the circle, and the fourth lands on it only by coincidence — the corners of a rectangle do, the corners of a general quadrilateral do not.
Concept
A circle is pinned down by three numbers: the two coordinates of its centre and its radius.
Each point you insist it passes through is one equation, so three points give three equations in three unknowns and, for non-collinear points, exactly one solution.
That solution is the circumcircle of the triangle the three points form, and its centre is where the perpendicular bisectors of the sides meet.
A fourth point adds a fourth equation to the same three unknowns. It can still be satisfied, but only by special sets of points — those are the ones called concyclic.
Non-collinear is the whole of the condition, not decoration. Three points on one line have no circumcircle at all, because the perpendicular bisectors of the two segments are parallel.
Key facts
- Exactly one circle passes through three non-collinear points.
- Its centre is the circumcentre, where the perpendicular bisectors of the sides of the triangle meet.
- Three collinear points lie on no circle.
- Four or more points share a circle only when they are concyclic, as the vertices of a rectangle are.
Study next
Common traps
- Reading the blank as the largest number of points a circle can pass through, which is unlimited
- Overlooking that the answer becomes no circle at all once the points are collinear
SSC uses this as a one-line definition check with no calculation, so the marks turn on knowing the word non-collinear.
Circle geometry is also asked in this paper at Quant Q.9, where two circles with centres 36 cm apart need a direct and a transverse common tangent.
Related PYQs
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