JEE · Mathematics
Recognise and construct circle equations, extract centre and radius safely, classify real, degenerate and nonreal loci, and use tangent, chord and parametric methods only where geometric conditions and official scope support them.
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In short
A circle is the locus of points at a fixed nonnegative distance from a centre. In x^2 + y^2 + 2gx + 2fy + c = 0, the centre is (-g, -f) and the radius squared is g^2 + f^2 - c.
A proper real circle requires that quantity to be positive; zero gives a point circle, and negative gives no real circle.
The official JEE documents define content scope. They do not publish chapter weightage, so none is asserted here.
Sources: JEE Main and JEE Advanced official syllabi, linked in the sources section below.
This is a readiness check, not a weightage or scoring-priority list.
Standardise -> compare coefficients -> compute r^2 -> classify -> choose tangent/intersection method -> validate distance.
Standardise the given equation, compare coefficients with the general form, compute r squared before taking a square root, classify the locus as a proper circle, a point circle, or no real circle, then choose the correct tangent or intersection method and validate through distance.
Complete the square; compute r^2 before r; check the sign before taking a square root.
For x^2 + y^2 + 2gx + 2fy + c = 0, the centre is (-g, -f) and radius squared is g^2 + f^2 - c. Computing r^2 first, then checking its sign, avoids taking a square root of a negative quantity before the locus is classified.
r^2 > 0 gives a proper circle; r^2 = 0 gives a point circle; r^2 < 0 gives no real circle.
Not every equation with equal x^2 and y^2 coefficients and no xy term after normalisation represents a real, nondegenerate circle. The quantity g^2 + f^2 - c must be checked: positive gives a proper circle with r > 0, zero gives a single point, and negative gives no real locus.
The diameter-endpoint equation requires distinct endpoints; the position expression gives inside, on, or outside.
Given diameter endpoints, (x - x1)(x - x2) + (y - y1)(y - y2) = 0 constructs the circle, provided the endpoints are distinct so the circle is nondegenerate. For a point (x0, y0) and a proper circle, the signed expression (x0 - h)^2 + (y0 - k)^2 - r^2 is negative inside, zero on, and positive outside the circle.
The tangent-at-a-point formula and the parametric form require the point or parameter to correspond to the actual circle; these are Advanced-explicit methods.
The tangent at P(x1, y1) on a circle, (x1 - h)(x - h) + (y1 - k)(y - k) = r^2, applies only when P lies on the circle; applying it to an off-circle point produces an incorrect line. The parametric point (h + r cos t, k + r sin t) applies to a proper circle after standard-form identification. These are listed explicitly for JEE Advanced 2026 and are not separately named in the JEE Main 2026 unit.
g = -3, f = 2, c = -12.
Centre (3, -2). r^2 = 9 + 4 + 12 = 25, so r = 5.
P = (3, 3) lies on the circle. Radius CP is vertical, so the tangent is y = 3.
Validation: the tangent is perpendicular to CP and passes through P.
Reversing the sign when reading the centre from the general form.
Execution errorWhy it happens
The centre is (-g, -f), not (g, f), for x^2 + y^2 + 2gx + 2fy + c = 0.
How it is corrected
Complete the square explicitly rather than reading off g and f without the sign flip.
Failing to check whether g^2 + f^2 - c is positive before proceeding.
Decision / selection errorWhy it happens
A negative value means no real circle exists, and zero means a point circle rather than a proper circle.
How it is corrected
Compute and check the sign of r^2 before taking a square root or applying tangent methods.
Treating r = 0 as a proper circle.
Decision / selection errorWhy it happens
A radius of zero reduces the locus to a single point, not a circle with tangent lines and chords.
How it is corrected
Classify r^2 = 0 as a point circle before applying any circle-specific method.
Using the tangent-at-a-point formula for a point not on the circle.
Decision / selection errorWhy it happens
That formula assumes the given point already lies on the circle.
How it is corrected
Verify the point satisfies the circle equation before applying the tangent formula.
Calling every quadratic equation with x^2 and y^2 terms a circle.
Knowledge gapWhy it happens
A circle requires equal coefficients for x^2 and y^2 and no xy term after normalisation.
How it is corrected
Check coefficient equality and the absence of an xy term before classifying an equation as a circle.
Presenting tangent, normal, chord or parametric circle methods as explicit Main 2026 detail.
Knowledge gapWhy it happens
These methods are listed explicitly for JEE Advanced 2026, not separately named in the JEE Main 2026 unit.
How it is corrected
Label tangent, normal, chord and parametric circle methods as Advanced-explicit scope.
FAQ
Straight answers about how Rank Sarthi fits into serious exam preparation.
For x^2 + y^2 + 2gx + 2fy + c = 0, the centre is (-g, -f) and the radius squared is g^2 + f^2 - c.
The quantity g^2 + f^2 - c must be positive for a proper real circle.
A radius of zero, where g^2 + f^2 - c equals zero, reduces the locus to a single point rather than a proper circle.
For distinct diameter endpoints (x1, y1) and (x2, y2), the circle equation is (x - x1)(x - x2) + (y - y1)(y - y2) = 0.
Tangent, normal, chord, parametric circle, and circle-line or circle-circle intersection constructions are listed explicitly for JEE Advanced 2026.
Circle equation, classification and tangent/intersection scope is verified against the official JEE Main 2026 and JEE Advanced 2026 syllabi only. Exact provenance only; no frequency or trend claims.
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