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CDS Conic Sections

Study Material — 5 PYQs (2020–2020) · Concept Notes · Shortcuts

CDS Conic Sections is a frequently tested subtopic — 5 previous year questions from 2020–2020 papers are included below with concept notes, key rules and shortcut tricks.

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2020–2020
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Previous Year Questions

CDS Conic Sections — Past Exam Questions

5 questions from actual CDS papers · all shown free · click option to reveal solution

Exam Q 12020Previous Year Pattern

The equation of the hyperbola with foci at (±5, 0) and the difference of distances from any point on the hyperbola to the two foci equal to 6 is:

Exam Q 22020Previous Year Pattern

The equation of the parabola with vertex at the origin, axis along the positive x-axis, and passing through the point (2, 4) is:

Exam Q 32020Previous Year Pattern

The foci of the ellipse 9x² + 16y² = 144 are located at:

Exam Q 42020Previous Year Pattern

The equation of the tangent to the ellipse x²/25 + y²/9 = 1 at the point (5, 0) is:

Exam Q 52020Previous Year Pattern

A point P moves such that its distance from the line x = 4 equals its distance from the point F(1, 0). The locus of P is:

Concept Notes

Conic Sections— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic

CORE CONCEPT A conic section is a curve you get when a plane cuts through a cone. Depending on the angle of the cut, you get four shapes: Circle, Ellipse, Parabola, or Hyperbola. These shapes appear everywhere in NDA Maths and understanding their standard equations is the key to solving problems fast.

KEY RULES / PROPERTIES Every conic section has a general second-degree equation: Ax² + Bxy + Cy² + Dx + Ey + F = 0. The value of the discriminant B² - 4AC tells you which conic it is.

- If B² - 4AC < 0 and A = C: Circle - If B² - 4AC < 0 and A ≠ C: Ellipse

- If B² - 4AC = 0: Parabola - If B² - 4AC > 0: Hyperbola

2.Formula Block

Memorise — at least one formula appears in every paper
Circle (centre at origin): x² + y² = r²
Circle (centre at h, k): (x - h)² + (y - k)² = r²
Parabola (opens right): y² = 4ax, Focus = (a, 0), Directrix: x = -a
Parabola (opens up): x² = 4ay, Focus = (0, a), Directrix: y = -a
Ellipse (horizontal): x²/a² + y²/b² = 1, where a > b
Eccentricity of Ellipse: e = c/a, where c² = a² - b², and 0 < e < 1
Hyperbola: x²/a² - y²/b² = 1
Eccentricity of Hyperbola: e > 1, c² = a² + b²
Rectangular Hyperbola: xy = c²

3.Exam Patterns

What examiners ask — read before attempting PYQs

NDA frequently asks: 1. Identify the conic from a given equation 2. Find the focus, directrix, or eccentricity 3. Find the equation of a conic given certain conditions 4.

Number of points of intersection between two conics SHORTCUT / TRICK Trick 1 — Eccentricity Memory Rule: Circle → e = 0 Ellipse → 0 < e < 1 Parabola → e = 1 Hyperbola → e > 1 Just remember: C-E-P-H with values 0, between 0-1, exactly 1, greater than 1. Trick 2 — For parabola y² = 4ax, the latus rectum length = 4a. For ellipse x²/a² + y²/b² = 1, latus rectum = 2b²/a. No need to derive — just memorise these directly.

4.Worked Example

Solve this step-by-step before moving on
1
Step 1

Compare y² = 12x with standard form y² = 4ax. So 4a = 12, which gives a = 3.

2
Step 2

Focus of y² = 4ax is at (a, 0). Focus = (3, 0)

3
Step 3

Directrix is x = -a. Directrix: x = -3

4
Step 4

Length of latus rectum = 4a = 4 × 3 = 12. Answer: Focus = (3, 0), Directrix: x = -3, Latus Rectum = 12.

5.Exam Traps

Common mistakes students make — avoid these

Students confuse the formulas for ellipse and hyperbola. In ellipse, c² = a² - b² (subtract). In hyperbola, c² = a² + b² (add).

Also, many forget that for ellipse, a > b always (a is under x² for horizontal ellipse). Always check this condition first.

Key Points to Remember

  • Four conic sections: Circle, Ellipse, Parabola, Hyperbola — formed by cutting a cone at different angles.
  • Discriminant rule: B² - 4AC < 0 → Circle or Ellipse; = 0 → Parabola; > 0 → Hyperbola.
  • Standard parabola y² = 4ax has Focus at (a, 0) and Directrix at x = -a.
  • Eccentricity: Circle e=0, Ellipse 0<e<1, Parabola e=1, Hyperbola e>1.
  • For ellipse x²/a² + y²/b² = 1: c² = a² - b² and eccentricity e = c/a.
  • For hyperbola x²/a² - y²/b² = 1: c² = a² + b² and eccentricity e = c/a > 1.
  • Latus rectum of parabola y² = 4ax = 4a; Latus rectum of ellipse = 2b²/a.
  • Rectangular hyperbola equation is xy = c², and its eccentricity = root 2 (approximately 1.414).

Exam-Specific Tips

  • For parabola y² = 4ax: Focus = (a, 0), Vertex = (0, 0), Directrix = x = -a, Latus Rectum = 4a.
  • Eccentricity of a circle is exactly 0; eccentricity of a parabola is exactly 1.
  • For ellipse x²/a² + y²/b² = 1 with a > b: Length of major axis = 2a, minor axis = 2b.
  • For hyperbola x²/a² - y²/b² = 1: The asymptotes are y = (b/a)x and y = -(b/a)x.
  • Eccentricity of rectangular hyperbola (xy = c²) is always equal to root 2.
  • The general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 represents a circle when B = 0 and A = C.
  • Sum of focal distances of any point on an ellipse = 2a (the length of the major axis).
  • Difference of focal distances of any point on a hyperbola = 2a (constant).
Practice MCQs

Conic Sections — Practice Questions

28graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 28

All MCQs →
Practice 1easy

The eccentricity of the hyperbola x²/25 − y²/9 = 1 is:

Practice 2easy

The eccentricity of the hyperbola 16x² - 9y² = 144 is:

Practice 3easy

The equation of the circle with center (−2, 3) and radius 5 is:

Practice 4easy

The eccentricity of the ellipse 9x² + 16y² = 144 is:

Practice 5easy

The length of the latus rectum of the hyperbola x²/25 − y²/9 = 1 is:

Practice 6easy

The equation of the circle with centre (−2, 3) and radius 5 is:

Practice 7easy

The foci of the ellipse 4x² + 9y² = 36 are located at:

Practice 8easy

The length of the latus rectum of the parabola y² = 12x is:

Practice 9medium

For the hyperbola x²/25 − y²/16 = 1, the length of the latus rectum is:

Practice 10medium

The equation of the tangent to the parabola y² = 12x at the point (3, 6) is:

Practice 11medium

The eccentricity of the hyperbola 16x² − 9y² = 144 is:

Practice 12medium

A circle passes through the points (1, 0), (0, 1), and (−1, 0). The equation of the circle is:

Practice 13medium

The equation of the tangent to the circle x² + y² − 4x − 6y + 9 = 0 at the point (2, 1) is:

Practice 14medium

The eccentricity of the ellipse 4x² + 9y² = 36 is:

Practice 15medium

The eccentricity of the hyperbola 16x² - 9y² = 144 is:

Practice 16medium

The length of the latus rectum of the ellipse x²/25 + y²/9 = 1 is:

Practice 17hard

An ellipse has the equation 9x² + 16y² = 144. A point P on the ellipse is such that the sum of its distances from the two foci is k times the length of the major axis. Find k.

Practice 18hard

A parabola has its vertex at the origin and focus at (0, 3). A line passing through the focus intersects the parabola at points P and Q. If the length of the chord PQ is 12, find the sum of the ordinates of P and Q.

Practice 19hard

A parabola has its vertex at the origin and its focus at (3, 0). A line passing through the focus intersects the parabola at points P and Q. If the length of the focal chord PQ is 12, find the sum of the ordinates of P and Q.

Practice 20hard

An ellipse has the equation 9x² + 16y² = 144. A point P on the ellipse is such that the sum of its distances from the two foci is k times the length of the major axis. If k = 1/2, find the eccentricity of the ellipse.

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60-Second Revision — Conic Sections

  • Remember: Eccentricity order — Circle(0) → Ellipse(0 to 1) → Parabola(1) → Hyperbola(>1). This alone can solve many MCQs.
  • Formula: Parabola y² = 4ax → match coefficient of x to find 4a, then a gives you focus (a,0) and directrix x = -a directly.
  • Formula: Ellipse uses c² = a² MINUS b²; Hyperbola uses c² = a² PLUS b². Never swap these.
  • Trap: In ellipse x²/a² + y²/b² = 1, always confirm a > b. If b > a, swap roles — major axis is along y-axis.
  • Remember: Latus Rectum of parabola = 4a; of ellipse = 2b²/a; of hyperbola = 2b²/a. Same formula for ellipse and hyperbola.
  • Trick: Use B² - 4AC to identify conic type quickly from a general equation without solving it fully.
  • Trap: Rectangular hyperbola xy = c² looks different from standard form — its eccentricity is root 2, not variable.
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