Conic sections are the curves you get when a plane slices a double cone — the circle, parabola, ellipse and hyperbola. In the NDA written exam this single chapter quietly delivers several marks every year, and the questions are highly formula-driven. Learn the standard equations once, and you can solve most problems by simply comparing and substituting.
Why conic sections matter for NDA
Coordinate geometry is one of the most scoring areas in NDA Mathematics because the answers come from fixed formulas rather than long reasoning. Circles and conics together contribute a steady handful of questions in every paper.
The good news for a Class 11-12 student with limited time: you do not need to derive anything in the exam. You need to recognise the form, pull out the constants, and apply the right formula. There is no proof-writing and no long calculation here; just pattern matching followed by a clean substitution. Two students with the same algebra skills can score very differently in this chapter purely based on whether the standard equations are at their fingertips, so a focused hour of memorising the forms below turns these questions into almost-free marks.
A conic is the locus of a point that moves so that its distance from a fixed point (the focus) bears a constant ratio (the eccentricity, e) to its distance from a fixed line (the directrix).
That ratio e is the single number that tells you which conic you are looking at, so we will keep coming back to it.
Eccentricity: the one number that names the curve
Every conic has an eccentricity e ≥ 0. The value of e instantly classifies the curve.
- Circle: e = 0
- Ellipse: 0 < e < 1
- Parabola: e = 1
- Hyperbola: e > 1
So a circle is really a special ellipse where the two foci have merged into the centre. A parabola sits exactly on the boundary at e = 1. This ladder — 0, between 0 and 1, exactly 1, more than 1 — is worth memorising cold, because NDA loves to ask "which of the following is a parabola/ellipse?" type questions.
If a question gives you a general second-degree equation Ax2 + Bxy + Cy2 + ... = 0 with no xy term (B = 0): equal coefficients of x2 and y2 → circle; same sign but unequal → ellipse; opposite signs → hyperbola; one square missing → parabola.
The circle and its standard equations
A circle is the set of all points at a fixed distance (the radius r) from a fixed point (the centre).
Centre (h, k), radius r:
(x − h)2 + (y − k)2 = r2
General form:
x2 + y2 + 2gx + 2fy + c = 0
with centre (−g, −f) and radius r = √(g2 + f2 − c).
Two quick truths flow from the general form. If centre is the origin, the equation reduces to x2 + y2 = r2. And the coefficients of x2 and y2 are always equal with no xy term — that is the fingerprint of a circle.
In the general form the centre is (−g, −f), not (g, f). And before reading g and f you must make the coefficient of x2 and y2 equal to 1. If the equation is 2x2 + 2y2 + ... = 0, divide the whole thing by 2 first.
Real circle, tangents and useful circle facts
For x2 + y2 + 2gx + 2fy + c = 0 to represent a real circle, we need g2 + f2 − c > 0. If it equals 0 the "circle" is a single point; if it is negative the circle is imaginary.
- Length of tangent from an external point (x1, y1) to S = 0 is √S1, where S1 = x12 + y12 + 2gx1 + 2fy1 + c.
- If S1 > 0 the point is outside, S1 = 0 on the circle, S1 < 0 inside.
- A line is a tangent when the perpendicular distance from the centre to the line equals the radius.
The diameter form is also handy: if (x1, y1) and (x2, y2) are ends of a diameter, the circle is (x − x1)(x − x2) + (y − y1)(y − y2) = 0.
The parabola (e = 1)
A parabola is the locus of a point equidistant from the focus and the directrix. The simplest standard form opens to the right.
For y2 = 4ax (a > 0):
- Vertex: (0, 0)
- Focus: (a, 0)
- Directrix: x = −a
- Axis: the x-axis (y = 0)
- Length of latus rectum: 4a
The other three orientations follow the same pattern: y2 = −4ax opens left, x2 = 4ay opens up, and x2 = −4ay opens down. The number multiplying the linear term is always 4a, so comparing instantly gives you a.
The latus rectum is the focal chord perpendicular to the axis. For any parabola its length is 4a, for an ellipse or hyperbola it is 2b2/a. NDA repeatedly asks for the length of the latus rectum, so keep these three values ready.
The ellipse (0 < e < 1)
An ellipse looks like a stretched circle. Take its standard form with the major axis along the x-axis.
For x2/a2 + y2/b2 = 1 with a > b:
- Major axis length 2a, minor axis length 2b
- Foci: (±ae, 0), where b2 = a2(1 − e2)
- Eccentricity: e = √(1 − b2/a2)
- Directrices: x = ±a/e
- Latus rectum length: 2b2/a
The defining property: for any point on the ellipse, the sum of distances to the two foci is constant and equals 2a. That constant-sum idea is the cleanest way to remember the ellipse.
The formula b2 = a2(1 − e2) only holds when a is the larger denominator. If b > a (major axis along y-axis), swap the roles: a2 = b2(1 − e2). Always put the bigger number under the axis that is longer.
The hyperbola (e > 1)
A hyperbola is two open branches. Its standard form looks like the ellipse but with a minus sign.
For x2/a2 − y2/b2 = 1:
- Vertices: (±a, 0), foci: (±ae, 0)
- Relation: b2 = a2(e2 − 1)
- Eccentricity: e = √(1 + b2/a2)
- Asymptotes: y = ±(b/a)x
- Latus rectum length: 2b2/a
Here the defining property uses a difference: the absolute difference of distances from any point to the two foci is constant and equals 2a. Note the sign flips inside the e formulas: ellipse has 1 − ..., hyperbola has 1 + ... (and e2 − 1 inside b2).
A rectangular (equilateral) hyperbola has a = b, which forces e = √2 and perpendicular asymptotes. xy = c2 is the rectangular hyperbola referred to its asymptotes.
Side-by-side: all four conics at a glance
Most NDA questions are solved by spotting which row below you are in. Keep this mental table sharp.
- Circle — e = 0; equal x2, y2 coefficients; centre + radius.
- Parabola — e = 1; only one variable squared; latus rectum 4a.
- Ellipse — 0 < e < 1; both squares, same sign, plus sign; sum of focal distances = 2a.
- Hyperbola — e > 1; both squares, minus sign; difference of focal distances = 2a.
For both ellipse and hyperbola the latus rectum is 2b2/a and the foci are at (±ae, 0). The only structural difference is the sign and which of (1 − e2) or (e2 − 1) appears. Lock that down and half the chapter is done.
A practical exam habit: when you see a second-degree equation, first scan the squared terms and their signs, then immediately decide circle, parabola, ellipse or hyperbola before doing any arithmetic. That single decision tells you which formula set in your head to open, and it stops you from blindly applying an ellipse formula to a hyperbola or the other way round.
Worked example: find the conic and its features
Identify the conic 9x2 + 25y2 = 225 and find its eccentricity, foci and length of latus rectum.
So the curve is an ellipse with e = 4/5, foci at (±4, 0), and latus rectum 18/5. Notice how every answer dropped out of three standard formulas once we reduced the equation to standard form. That is the whole skill.
Common mistakes to avoid
Conics are easy marks, but careless errors throw them away. Watch for these.
- Forgetting to divide so the right side becomes 1 before reading a2 and b2.
- Mixing up the ellipse relation b2 = a2(1 − e2) with the hyperbola relation b2 = a2(e2 − 1).
- Reading the circle centre as (g, f) instead of (−g, −f).
- Assuming a is always under x2; for a vertical ellipse the larger denominator is under y2.
Before applying any focus or directrix formula, write the equation in clean standard form and state which axis is major/transverse. Thirty seconds of setup prevents the most common sign slips.
Previous-year style question
Q. The eccentricity of the hyperbola x2/16 − y2/9 = 1 is:
Answer: Here a2 = 16 and b2 = 9, so for a hyperbola e = √(1 + b2/a2) = √(1 + 9/16) = √(25/16) = 5/4. So e = 5/4 = 1.25, which is correctly greater than 1 as expected for a hyperbola.
Q. Find the centre and radius of the circle x2 + y2 − 6x + 4y − 12 = 0.
Answer: Comparing with x2 + y2 + 2gx + 2fy + c = 0 gives 2g = −6, 2f = 4, c = −12, so g = −3, f = 2. Centre = (−g, −f) = (3, −2). Radius = √(g2 + f2 − c) = √(9 + 4 + 12) = √25 = 5.
Quick revision
- Eccentricity ladder: circle 0, ellipse 0–1, parabola 1, hyperbola >1.
- Circle: (x−h)2+(y−k)2=r2; general centre (−g,−f), r=√(g2+f2−c).
- Parabola y2=4ax: focus (a,0), directrix x=−a, latus rectum 4a.
- Ellipse: foci (±ae,0), b2=a2(1−e2), LR = 2b2/a, sum of focal distances = 2a.
- Hyperbola: foci (±ae,0), b2=a2(e2−1), asymptotes y=±(b/a)x, difference of focal distances = 2a.
- Always reduce to standard form (right side = 1) before reading constants.
Drill ten mixed PYQs identifying the conic and its eccentricity, and this chapter becomes guaranteed marks in your NDA Maths paper with The Cavalier.
Frequently asked questions
How are circles and conic sections weighted in the NDA Maths paper?
Coordinate geometry, including circles and conics, contributes a steady set of questions in every NDA Mathematics paper. They are formula-driven and quick to solve, making them among the most scoring topics if your standard forms are memorised.
What is the fastest way to tell which conic an equation represents?
Check the squared terms. Equal coefficients of x-squared and y-squared with no xy term means a circle; one variable squared means a parabola; both squared with the same sign means an ellipse; both squared with opposite signs means a hyperbola.
What is the difference between the ellipse and hyperbola formulas for eccentricity?
For an ellipse, b-squared = a-squared times (1 minus e-squared) and e is less than 1. For a hyperbola, b-squared = a-squared times (e-squared minus 1) and e is greater than 1. The sign inside the bracket flips between the two.
What is the length of the latus rectum for each conic?
For a parabola y-squared = 4ax it is 4a. For both an ellipse and a hyperbola in standard form it is 2 times b-squared divided by a. These appear frequently in NDA questions, so keep them memorised.
How do I find the centre of a circle from its general equation?
Write it as x-squared + y-squared + 2gx + 2fy + c = 0, ensuring the coefficient of the squared terms is 1. Then the centre is (minus g, minus f) and the radius is the square root of (g-squared + f-squared minus c).
Related NDA Maths topics
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