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Circle questions often look new each time, but they draw on a small set of results. If you know how to read the centre and radius from the general equation, how to compare a distance with the radius, and three short equations written in S and T notation, you can handle most tangent, chord and family problems. This post works through one circle from start to finish so that every number can be checked against the others.
What does the syllabus cover?
The JEE (Main) 2026 syllabus lists the standard and general forms of a circle, its radius and centre, the circle with given endpoints of a diameter, and the points of intersection of a line and a circle with centre at the origin. Tangents, normals and chords, the parametric form and the "equation of a circle through the points of intersection of two circles and those of a circle and a straight line" are named in the JEE (Advanced) 2026 syllabus. The CBSE 2026-27 curriculum asks Class 11 students for the standard equation of a circle, which NCERT covers in its chapter on conic sections. Check the documents for your exam year.
How do you read the centre and radius?
Write the circle as S = x² + y² + 2gx + 2fy + c = 0. The centre is (−g, −f) and the radius is √(g² + f² − c). If g² + f² − c is negative, there is no real circle.
Example. S: x² + y² − 6x + 4y − 12 = 0. Here 2g = −6 and 2f = 4, so g = −3 and f = 2. The centre is (3, −2) and the radius is √(9 + 4 + 12) = √25 = 5.
We will use this circle for every example below. Two pieces of notation help:
S₁ means S evaluated at a point (x₁, y₁).
T means xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c.
When is a line tangent to a circle?
A line is tangent when its distance from the centre equals the radius. It cuts the circle at two points when the distance is smaller, and misses it when larger.
Example. For which k is 3x − 4y + k = 0 tangent to S? The distance from (3, −2) is |9 + 8 + k|/5. Set it equal to 5: |17 + k| = 25, so k = 8 or k = −42. There are two answers because there are two parallel tangents, one on each side of the circle.
How do you write the tangent at a point on the circle?
Use T = 0. The point (6, 2) is on S, because (6 − 3)² + (2 + 2)² = 9 + 16 = 25.
T = 6x + 2y − 3(x + 6) + 2(y + 2) − 12 = 6x + 2y − 3x − 18 + 2y + 4 − 12 = 3x + 4y − 26.
So the tangent is 3x + 4y − 26 = 0. Two checks: its distance from the centre is |9 − 8 − 26|/5 = 25/5 = 5, the radius; and its normal direction (3, 4) points along the radius from (3, −2) to (6, 2).
How long is the tangent from an outside point?
The length of the tangent from (x₁, y₁) is √S₁, as long as the x² and y² coefficients are 1.
Example. From (10, 3): S₁ = 100 + 9 − 60 + 12 − 12 = 49, so the length is 7. Check with geometry: the distance from (10, 3) to the centre is √(49 + 25) = √74, and √(74 − 25) = √49 = 7.
How do you find a chord with a given midpoint?
The chord with midpoint (x₁, y₁) is T = S₁.
Example. Midpoint (4, −1). T = 4x − y − 3(x + 4) + 2(y − 1) − 12 = x + y − 26. S₁ = 16 + 1 − 24 − 4 − 12 = −23. So x + y − 26 = −23, which gives x + y − 3 = 0. Check: the line from the centre (3, −2) to (4, −1) has direction (1, 1), and the chord has direction (1, −1), so they are perpendicular, as they must be.
What is a family of circles?
Every circle through the intersection points of S = 0 and a line L = 0 can be written S + λL = 0. For two circles, S₁ + λS₂ = 0 works the same way (here S₁ and S₂ name two circles, not values at a point), and λ = −1 gives the line of their common chord.
Example. Find the circle through the intersection points of x² + y² = 9 and x + y = 1 that passes through the origin. Write x² + y² − 9 + λ(x + y − 1) = 0. At (0, 0): −9 − λ = 0, so λ = −9, and the circle is x² + y² − 9x − 9y = 0. Check: at any intersection point x² + y² = 9 and x + y = 1, so 9 − 9(1) = 0.
Which mistakes cost the most marks?
Forgetting to halve. In x² + y² − 6x + 4y − 12 = 0, g is −3, not −6.
Sign of the centre. The centre is (−g, −f), so −6x means the centre has x = +3.
Using T = 0 at a point not on the circle. Check S₁ = 0 first; for an outside point, T = 0 is the chord of contact, not a tangent.
One tangent instead of two when a slope or a parallel condition is given.
How should you study these results?
Do not learn T = S₁ as a rule to apply blindly. A review of worked-example research by Atkinson, Derry, Renkl and Wortham (2000) recommends several examples for each problem type, varied formats within a type, practice problems placed close to the examples, and encouraging learners to explain the examples to themselves.
One circle, five questions (weekly): take any circle and find its centre, a tangent at a point, a tangent length, a chord with given midpoint and one family member, as above. Each result checks the others.
Explain each step aloud: say why the distance equals the radius, or why the chord is perpendicular to the line from the centre.
Mix with lines: our guide to straight lines in coordinate geometry covers the distance formula used here.
If a family-of-circles question will not come out and you cannot see which step went wrong, the Fermi Problem Library has a step-by-step solver that explains the reasoning, so you can find the step where your working split from it.
Tonight, pick a new circle and work through all five questions on it, checking each answer against another.
