Circles and Conic Sections

What This Topic Covers

The NDA Mathematics syllabus groups Circles and Conic Sections under Analytical Geometry (two dimensions). The examinable content includes:

• Standard and general equation of a circle; conditions for the general second-degree equation to represent a circle.
• Circle touching coordinate axes; intercepts on axes; distance between centres of two circles.
• Parabola — standard forms, vertex, focus, directrix, axis, latus rectum, focal distance.
• Ellipse — standard form, major and minor axes, foci, eccentricity (0 < e < 1), sum of focal distances, latus rectum.
• Hyperbola — standard form, transverse and conjugate axes, foci, eccentricity (e > 1), difference of focal distances, rectangular hyperbola.
• Identification of the conic from a general second-degree equation.

This topic links directly to Straight Lines and Cartesian System (tangent conditions, chord problems) and feeds into Three-Dimensional Geometry (analogous quadric surfaces).

Exam Pattern & Weightage

The table below is built from the PYQ files. Each year typically holds one NDA I and one NDA II paper; the count column shows questions from both sittings combined.

The combined total runs to well over 100 PYQs. The trend since 2022 shows an increase — 2023 produced 11 questions and 2025 produced 9. Never skip this topic in your final revision.

Core Concepts

General Second-Degree Equation — Identifying the Conic

Every conic (and degenerate pair of lines) springs from one mother equation:

Compute the discriminant \(\Delta = abc + 2fgh - af^2 - bg^2 - ch^2\). Then read off the curve from \(h^2\) versus \(ab\):

Circle — Standard and General Form

A circle with centre (h, k) and radius r has the standard equation:

Expanding gives the general second-degree form:

The general second-degree equation \(ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0\) represents a circle when \(a = b\) and \(h = 0\). Both conditions must hold simultaneously — this was directly tested in 2010-II and 2011-I.

This form appears when the endpoints of a diameter are given. It was tested in 2018-II.

Circle Touching the Axes

When a circle touches both coordinate axes, its centre is \((r, r)\), \((-r, r)\), \((r, -r)\), or \((-r, -r)\) where \(r\) is the radius. The standard template equation is:

To find \(r\), use any additional constraint — e.g., the centre lies on a given line. In 2010-II: centre on \(x + y = 4\) gives \(r + r = 4\), so \(r = 2\), producing \(x^2 + y^2 - 4x - 4y + 4 = 0\).

Parabola — Standard Forms

The four standard parabolas and their key features:

Focal distance of any point \(P(x_1, y_1)\) on \(y^2 = 4ax\) equals \(a + x_1\). This was derived and tested in 2011-II. Latus rectum length \(= 4a\) for all four forms.

One parameter \(t\) gives every point on the parabola — useful when a chord, tangent, or focal-chord question hands you two parameters \(t_1, t_2\) instead of coordinates.

Ellipse — Standard Form and Eccentricity

The sum of focal distances of any point on the ellipse equals the length of the major axis (\(2a\)). This definition appeared in 2011-I, 2012-II, 2015-I, 2019-I, 2020-I, 2023-I, and 2024-II — memorise it cold.

When the latus rectum equals half the minor axis: \(2b^2/a = b\), so \(a = 2b\). Then \(e = \sqrt{1 - b^2/a^2} = \sqrt{1 - 1/4} = \sqrt{3}/2\). This result was tested in both 2010-II and 2012-I.

Hyperbola — Standard Form and Eccentricity

The only difference between the two formulas is the sign under the square root — a single character that decides the entire conic. Confirm which curve you have before plugging in.

Eccentricity Summary

This classification was directly asked in NDA 2021-I (statement-type question) with all three non-trivial conditions confirmed correct.

Worked Examples

Example 1 — Circle Touching Both Axes (NDA 2010-II)

Find the equation of the circle which touches both axes and has its centre on the line \(x + y = 4\).

• A circle touching both axes in the first quadrant has centre \((r, r)\) and equation $$x^2 + y^2 - 2rx - 2ry + r^2 = 0$$
• Centre \((r, r)\) must lie on \(x + y = 4\), so \(r + r = 4\), giving \(r = 2\).
• Substitute \(r = 2\): $$x^2 + y^2 - 4x - 4y + 4 = 0$$ This is option (b) — correct answer.

Example 2 — Point Inside a Circle (NDA 2013-I)

Which point lies inside a circle of radius 6 and centre \((3, 5)\)?

• A point \((x_0, y_0)\) is inside the circle if \((x_0 - h)^2 + (y_0 - k)^2 < r^2\).
• Test \((0, 1)\): \((0 - 3)^2 + (1 - 5)^2 = 9 + 16 = 25 < 36\). Inside the circle.
• Test \((2, -1)\): \((2-3)^2 + (-1-5)^2 = 1 + 36 = 37 > 36\). Outside.
• Answer: \((0, 1)\) lies inside the circle.

Example 3 — Eccentricity of an Ellipse (NDA 2010-II, 2012-I)

If the latus rectum of an ellipse equals half its minor axis, find the eccentricity.

• Latus rectum \(= 2b^2/a\); minor axis \(= 2b\). Condition: \(2b^2/a = b\), so \(a = 2b\).
• $$e^2 = 1 - \frac{b^2}{a^2} = 1 - \frac{b^2}{4b^2} = 1 - \frac{1}{4} = \frac{3}{4}$$
• Therefore \(e = \sqrt{3}/2\). Answer confirmed in answer key for both 2010-II and 2012-I.

Example 4 — Focal Distance on a Parabola (NDA 2024-I)

In the parabola \(y^2 = 8x\), the focal distance of point P is 8 units. Verify whether P can be \((6, 4\sqrt{3})\).

• Compare \(y^2 = 8x\) with \(y^2 = 4ax\): \(4a = 8\), so \(a = 2\).
• Focal distance formula \(= a + x_1\). For \(P = (6, 4\sqrt{3})\): focal distance \(= 2 + 6 = 8\). Matches the given condition. Statement A is correct.
• Perpendicular distance of P from directrix \(x = -2\): distance \(= 6 - (-2) = 8\). Statement B is also correct.
• Answer: Both A and B are correct.

Example 5 — Foci of an Ellipse and Sum of Distances (NDA 2024-II)

For the ellipse \(4x^2 + 9y^2 = 1\), foci are Q and R. For any point P on the ellipse, find \(PQ + PR\).

• Rewrite: $$\frac{x^2}{1/4} + \frac{y^2}{1/9} = 1$$ so \(a^2 = 1/4\), \(b^2 = 1/9\). Since \(a^2 > b^2\), major axis is along x-axis, \(a = 1/2\).
• Sum of focal distances of any point on an ellipse \(= 2a = 2 \cdot (1/2) = 1\).
• Answer: \(PQ + PR = 1\).

Example 6 — Centre and Radius from Non-Unit Coefficients

Find the centre and radius of the circle \(2x^2 + 2y^2 - 8x + 12y + 8 = 0\).

• The general-form formulas require the coefficients of \(x^2\) and \(y^2\) to be 1. Divide the whole equation by 2: $$x^2 + y^2 - 4x + 6y + 4 = 0$$
• Match with \(x^2 + y^2 + 2gx + 2fy + c = 0\): \(2g = -4 \Rightarrow g = -2\); \(2f = 6 \Rightarrow f = 3\); \(c = 4\).
• Centre \(=(-g,-f)=(2,-3)\); radius \(r=\sqrt{g^2+f^2-c}=\sqrt{4+9-4}=3\).

Example 7 — Eccentricity from a Non-Standard Ellipse

Find the eccentricity of the ellipse \(9x^2 + 16y^2 = 144\).

• Divide by 144 to get the standard form: $$\frac{x^2}{16} + \frac{y^2}{9} = 1$$
• Here \(a^2 = 16\), \(b^2 = 9\). Since \(a > b\), the major axis lies along the x-axis.
• Apply \(e = \sqrt{1 - b^2/a^2} = \sqrt{1 - 9/16} = \sqrt{7}/4\).

Example 8 — Position of a Point Relative to a Circle

Does the point (1, 2) lie inside, on, or outside the circle \(x^2 + y^2 - 4x - 6y + 4 = 0\)?

• Compute \(S_1\) by plugging the point into the LHS: \(S_1 = 1 + 4 - 4 - 12 + 4 = -7\).
• Sign rule: \(S_1 < 0\) means the point is inside; \(S_1 = 0\) means on the curve; \(S_1 > 0\) means outside.
• Here \(S_1 = -7 < 0\), so (1, 2) lies inside the circle.

Exam Shortcuts (Pro-Tips)

Circles and conics reward formula recognition over fresh derivation. The five shortcuts below convert what looks like a 2-minute derivation into a 15-second mental check. Every one of them has appeared in an NDA paper between 2010 and 2025.

Shortcut 1 — Eccentricity Classifier

Read the eccentricity, name the curve. NDA 2021-I asked this as a four-option statement question — full marks for memorising one row.

Bonus: rectangular hyperbola (\(a=b\)) gives \(e=\sqrt{2}\) instantly — tested NDA 2016-II.

Shortcut 2 — Parabola Focus/Directrix by Inspection

For any parabola in standard form, you do not need to derive — read the focus and directrix off the coefficient of 4a:

The sign of the right-hand side decides direction. \(y^2 = 8x\) means \(4a = 8\), so \(a = 2\), focus \((2,0)\), directrix \(x=-2\), latus rectum 8. Focal distance of any point on it = \(a + x_1\).

Shortcut 3 — Ellipse and Hyperbola b² Relations

Eccentricity questions on ellipse/hyperbola collapse to one line if you remember which sign sits between a² and b²:

From these: ellipse \(e = \sqrt{1 - b^2/a^2}\), hyperbola \(e = \sqrt{1 + b^2/a^2}\). The single sign difference is the most-tested trap in this topic.

Shortcut 4 — Position of a Point (S₁ Sign Test)

To check whether \((x_1, y_1)\) lies inside, on, or outside any conic \(S=0\), plug the point into the LHS and read the sign of the result \(S_1\):

No discriminants, no distance formulas. NDA 2013-I used this exact template for a circle.

Shortcut 5 — Tangency Conditions (Memorise These)

A line \(y = mx + c\) touches a standard conic only when \(c\) satisfies a fixed condition — memorise the three lines below and tangent problems become one-step substitutions:

For tangent at a given point on the conic, use the T = 0 rule: replace \(x^2 \to xx_1\), \(y^2 \to yy_1\), \(x \to (x+x_1)/2\), \(y \to (y+y_1)/2\). The result is the tangent line — no calculus needed.

Common Question Patterns

Pattern 1 — Condition for General Equation to Represent a Circle

The question gives \(ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0\) and asks the condition for it to be a circle. The answer is always: \(a = b\) and \(h = 0\). Tested in 2010-II and 2011-I with slight wording variations.

Pattern 2 — Radius from General Equation

Convert the general form to standard form by completing the square. For \(4x^2 + 4y^2 - 20x + 12y - 15 = 0\) (tested 2021-I): divide by 4 first to get \(x^2 + y^2 - 5x + 3y - 15/4 = 0\). Then \(g = -5/2\), \(f = 3/2\), \(c = -15/4\). Radius \(= \sqrt{25/4 + 9/4 + 15/4} = \sqrt{49/4} = 7/2 = 3.5\) units.

Pattern 3 — Two Circles Intersecting at Two Distinct Points

If two circles with radii \(r_1\) and \(r_2\) and centre distance \(d\) intersect at two distinct points, then \(|r_1 - r_2| < d < r_1 + r_2\). This gives a range for an unknown radius \(r\). Tested in 2016-I (\(2 < r < 8\)) and 2017-I (\(2 < r < 8\) again for a similar configuration).

Pattern 4 — Sum or Difference of Focal Distances

For an ellipse, sum \(= 2a\) (major axis length). For a hyperbola, difference \(= 2a\) (transverse axis length). NDA tests this as a recall question or as a back-calculation. You will see it almost every year in one form or another.

Pattern 5 — Identifying the Conic from a Given Equation

The equation \(2x^2 - 3y^2 - 6 = 0\) (tested 2019-I) rewrites as \(\frac{x^2}{3} - \frac{y^2}{2} = 1\) — a hyperbola. The equation \(x^2 + 3y = 0\) (tested 2025-II) rewrites as \(x^2 = -3y\), a downward parabola. Always rearrange to match a standard form before labelling.

Pattern 6 — Tangent to a Circle

The distance from the centre of a circle to a tangent line equals the radius. For the line \(x = y + 2\) (or \(x - y - 2 = 0\)) touching \(4(x^2 + y^2) = r^2\) (tested 2015-II), centre is \((0,0)\) and radius is \(r/2\). Distance \(= |0 - 0 - 2|/\sqrt{2} = 2/\sqrt{2} = \sqrt{2} = r/2\), giving \(r = 2\sqrt{2}\).

Preparation Strategy

Week 1 — Circles (Formula Drill)

Write out the standard form, general form, and diameter form from memory every day. Practice converting general equations to standard form (completing the square). Do all 37 circle PYQs from the file in timed 2-minute slots. Focus on: condition for circle, radius calculation, circle touching axes, intercepts on axes.

Week 2 — Parabola and Ellipse

Memorise all four parabola standard forms with their focus and directrix. Derive the focal distance formula \(a + x_1\) at least once by hand so you understand why it works. For ellipses, practise converting non-standard forms (like \(4x^2 + 9y^2 = 1\)) to the \(a^2/b^2\) form immediately. The sum-of-focal-distances property is pure recall — just remember \(2a\).

Week 3 — Hyperbola and Mixed Practice

Hyperbola questions often come with a parametric point such as \((3 \tan \theta, 2 \sec \theta)\) (tested 2021-I). Substitute into \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) to identify \(a\) and \(b\), then use \(e = \sqrt{1 + b^2/a^2}\). For rectangular hyperbola, \(a = b\) gives \(e = \sqrt{2}\). Run mixed sets of 10 questions (circles + all three conics) in 15 minutes — that is the NDA pace.

Revision Day — Eccentricity Sheet

One day before the exam, review nothing else from this topic except the eccentricity table and the intercept formulas. These are the easiest marks to lose through formula confusion under pressure.

For further coordinate geometry work, see Straight Lines and Cartesian System and Three-Dimensional Geometry. For topics that integrate with conics through calculus, see Limits, Continuity and Differentiability. Also useful: Vector Algebra (position vectors, locus problems) and NDA Maths — Full Subject Index.

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