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CBSE Class 11 Mathematics Chapter 10 Conic Sections Worksheet with Answers

Conic Sections form a critical chapter in CBSE Class 11 Mathematics, laying the foundation for coordinate geometry problems in both board exams and competitive tests like JEE. This worksheet systematically covers circle, parabola, ellipse, and hyperbola through six graded sections. Suggested time: 90 minutes. Difficulty: Intermediate. Attempt all sections honestly before checking the answer key.

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Key takeaways

✓Complete 90-minute worksheet covering all four conic sections tested in CBSE Class 11 board exams: circle, parabola, ellipse, and hyperbola.
✓Six varied sections include 6 MCQs, 5 fill-in-the-blanks, matching pairs, 5 short-answer questions, 3 HOTS long-answer problems, and 1 case study.
✓Answer key provides step-by-step solutions with formula references, helping students identify mistakes and understand methodology.
✓Intermediate difficulty level aligns with NCERT Class 11 Mathematics Chapter 10 terminology and recent CBSE examination patterns.
✓Questions test identification of conic type, standard form conversions, foci and directrix calculations, eccentricity, and real-world applications.
✓Printable format allows offline practice—students can attempt questions on paper, then verify using the detailed answer section.
✓Ideal for weekly revision, pre-exam drill, or as homework assignment for teachers following the CBSE Class 11 Mathematics syllabus.

Quick Chapter Recap: Conic Sections in NCERT Class 11 Mathematics

Chapter 10 of NCERT Class 11 Mathematics introduces conic sections as curves obtained by intersecting a double-napped right circular cone with a plane at different angles. When the plane cuts the cone parallel to its base, you get a circle. A cut parallel to a generator yields a parabola. An oblique slice that does not pass through the base produces an ellipse, and a vertical cut through both nappes gives a hyperbola. Each conic has a standard equation and geometric properties. A circle has equation (x−h)² + (y−k)² = r², centered at (h,k) with radius r. A parabola with vertex at the origin and focus on the x-axis takes the form y² = 4ax, where 'a' is the distance from vertex to focus. An ellipse centered at the origin with semi-major axis 'a' and semi-minor axis 'b' has equation x²/a² + y²/b² = 1 (a>b for horizontal major axis) or x²/b² + y²/a² = 1 (a>b for vertical major axis). A hyperbola centered at the origin opening left-right has equation x²/a² − y²/b² = 1, with foci along the x-axis. Eccentricity 'e' characterizes each conic: e=0 for circle, e=1 for parabola, 0<e<1 for ellipse, e>1 for hyperbola. Understanding these definitions, standard forms, foci, directrices, and the latus-rectum is essential for solving both theory and application problems in the CBSE board exams.

Circle: equation (x−h)² + (y−k)² = r², eccentricity e = 0
Parabola: standard form y²=4ax or x²=4ay, eccentricity e = 1, focus at (a,0) or (0,a)
Ellipse: x²/a² + y²/b² = 1, eccentricity e = √(1−b²/a²), 0 < e < 1
Hyperbola: x²/a² − y²/b² = 1, eccentricity e = √(1+b²/a²), e > 1
Latus-rectum is the chord through focus perpendicular to the major axis

Section A: Multiple Choice Questions (1 mark each)

Multiple-choice questions test your ability to quickly identify conic types, apply standard formulas, and perform algebraic manipulations. Each question below has four options; select the one correct answer. These six MCQs mirror the style seen in CBSE Class 11 term-end exams and sample papers released by the board. Pay close attention to signs, coefficients, and the placement of variables. In recent CBSE papers, MCQs on conic sections often ask for the focus coordinates, the value of eccentricity, or the type of conic from a given general equation. Work through each problem methodically: expand or complete the square if the equation is not in standard form, identify the conic by comparing coefficients of x² and y², then match with the provided options. Remember that a circle has equal coefficients for x² and y² with the same sign, a parabola has only one squared term, an ellipse has both squared terms with the same sign but different denominators, and a hyperbola has opposite signs. Time management is key—allocate roughly 1 minute per MCQ. If stuck, eliminate obviously incorrect options and make an educated guess, then revisit during review time.

Q1. The equation x² + y² − 6x + 4y − 12 = 0 represents which conic? (A) Circle (B) Parabola (C) Ellipse (D) Hyperbola
Q2. For the parabola y² = 16x, the coordinates of the focus are: (A) (4, 0) (B) (0, 4) (C) (−4, 0) (D) (0, −4)
Q3. The eccentricity of the ellipse 9x² + 16y² = 144 is: (A) √7/4 (B) √7/3 (C) 3/4 (D) 4/3
Q4. The length of the latus-rectum of the parabola x² = −20y is: (A) 5 (B) 10 (C) 20 (D) 40
Q5. Which of the following represents a hyperbola? (A) x²/9 + y²/16 = 1 (B) x²/9 − y²/16 = 1 (C) x² + y² = 25 (D) y² = 8x
Q6. The distance between the foci of the ellipse x²/25 + y²/9 = 1 is: (A) 4 (B) 8 (C) 6 (D) 10

Section B: Fill in the Blanks (1 mark each)

Fill-in-the-blank questions require precise recall of definitions, standard forms, and key results from NCERT Class 11 Mathematics Chapter 10. Each blank tests a specific concept—such as the formula for eccentricity, the coordinate of a vertex, or the equation of a directrix. These five items are designed to be completed in under 5 minutes total, so clarity and speed matter. Write your answer in the exact form requested: if the question asks for coordinates, enclose them in parentheses; if it asks for an equation, write it in standard form. Common pitfalls include sign errors and forgetting to simplify square roots. For instance, when asked for the eccentricity of an ellipse, students sometimes confuse the semi-major and semi-minor axes. Always double-check which axis is longer. In CBSE marking schemes, even a small algebraic slip can cost the mark, so verify your arithmetic before moving on. These blanks often appear in Section A of the board paper, contributing to the foundational 5–6 marks reserved for direct formula-based recall. Mastery here builds confidence for the more involved short-answer and long-answer sections that follow.

Q7. The standard equation of a circle with center (3, −2) and radius 5 is __________.
Q8. For the parabola y² = 4ax, the length of the latus-rectum is __________.
Q9. The eccentricity of a rectangular hyperbola is __________.
Q10. If the foci of an ellipse are (±4, 0) and the length of the major axis is 10, then the equation of the ellipse is __________.
Q11. The vertex of the parabola (y−2)² = 8(x+1) is __________.

Section C: Match the Following or True/False (1 mark each)

This section offers a mixed format to test conceptual clarity and the ability to connect definitions with examples. In CBSE Class 11 Mathematics examinations, match-the-following and true/false items often appear together as a single subsection worth 4–5 marks. For the matching exercise below, carefully read each conic property in Column A and find its corresponding equation or value in Column B. Write the correct letter–number pair in your answer sheet. For true/false statements, justify your answer in one sentence—CBSE marking schemes sometimes award half-marks for correct reasoning even if the T/F choice is wrong. Common traps include statements that are almost true but contain a subtle error, such as swapping semi-major and semi-minor axes or confusing the focus of a parabola with its directrix. Always refer back to the NCERT definitions: a parabola has exactly one focus and one directrix; an ellipse has two foci and two directrices; a hyperbola also has two foci and two directrices but with e > 1. Spend about 6–8 minutes on this section, ensuring you do not rush and make careless mistakes. These questions are designed to reinforce the logical structure of conic sections rather than test computational skill.

Match the Following: Column A — (i) Circle, (ii) Parabola, (iii) Ellipse, (iv) Hyperbola; Column B — (a) e = 1, (b) e > 1, (c) e = 0, (d) 0 < e < 1
Q12. Statement: The eccentricity of every circle is zero. True / False
Q13. Statement: A parabola has two foci. True / False
Q14. Statement: For the hyperbola x²/a² − y²/b² = 1, the foci lie on the y-axis. True / False
Q15. Statement: The latus-rectum of the ellipse x²/16 + y²/9 = 1 is 9/2. True / False

Section D: Short Answer Questions (2 or 3 marks each)

Short-answer questions in CBSE Class 11 Mathematics typically carry 2 or 3 marks and require you to show working across three to five clear steps. This section contains five questions that cover finding the equation of a conic given certain conditions, determining foci or vertices, calculating eccentricity, and converting a general second-degree equation into standard form by completing the square. Each question is designed to be solved in 4–5 minutes. Start by writing down the relevant formula or definition from NCERT Chapter 10, substitute the given values carefully, simplify algebraically, and box your final answer. For example, if asked to find the focus and directrix of a parabola, first rewrite the equation in the form y²=4ax or x²=4ay, identify 'a', then use the standard results: focus is (a,0) and directrix is x=−a for y²=4ax. Show every substitution step—CBSE examiners award partial credit for correct method even if the final answer contains a minor arithmetic error. These questions test your fluency with the standard forms and your ability to manipulate equations confidently. Practice these regularly, as they form the backbone of the 12–15 marks allocated to conic sections in the board paper.

Q16. Find the coordinates of the focus, vertex, and the equation of the directrix for the parabola y² = 12x. (3 marks)
Q17. Determine the center, vertices, and foci of the ellipse 4x² + 9y² = 36. (3 marks)
Q18. Find the eccentricity and the coordinates of the foci for the hyperbola 16x² − 9y² = 144. (3 marks)
Q19. Convert the equation x² + y² + 4x − 6y − 12 = 0 into standard form and identify the conic, its center, and radius. (2 marks)
Q20. A parabola has vertex at the origin and focus at (0, 3). Find its equation. (2 marks)

Section E: Long Answer / HOTS Questions (4 or 5 marks each)

Long-answer questions assess higher-order thinking skills—your ability to synthesize multiple concepts, apply coordinate geometry techniques, and solve multi-step problems involving conic sections. These three questions are designed to challenge students preparing for competitive exams like JEE as well as CBSE board toppers aiming for full marks. Each question should take 8–10 minutes and will test your command over derivations, parametric forms, tangent and normal equations, or applications of the distance formula combined with conic definitions. For instance, you may be asked to derive the equation of a tangent to an ellipse at a given point, or to prove that a certain locus is a parabola. Structure your answer clearly: state the given information, outline your approach, perform algebraic manipulations step-by-step, and conclude with a boxed final answer or statement. CBSE marking schemes reward logical flow and correct use of mathematical notation. Always define variables, label diagrams if provided, and cross-check units or dimensions. These HOTS questions often appear as Question 28 or 29 in the board paper, carrying significant weight. Mastering them requires not just formula recall but deep conceptual clarity on why each conic behaves the way it does, how eccentricity controls shape, and how geometric properties translate into algebra.

Q21. Derive the equation of the ellipse in standard form x²/a² + y²/b² = 1, given that the sum of distances from any point on the ellipse to the two foci is constant and equal to 2a. (5 marks)
Q22. A point moves such that its distance from the point (3, 0) is always twice its distance from the line x = 3/4. Show that the locus of the point is an ellipse and find its equation. (4 marks)
Q23. Find the equations of the tangent and normal to the parabola y² = 16x at the point (1, 4). (4 marks)

Section F: Case Study Question (4 marks)

Case-study questions have become a staple in recent CBSE Class 11 and 12 board papers, integrating real-world scenarios with mathematical modeling. This type of question presents a brief narrative—often involving architecture, physics, or engineering—and asks you to apply your knowledge of conic sections to answer sub-questions worth 1 mark each. The case study below involves a parabolic reflector, a common application of parabolas in satellite dishes and car headlights. Read the passage carefully, extract the given data, and use the standard properties of a parabola (focus, directrix, latus-rectum) to answer the four sub-parts. Each sub-question is designed to be concise, but together they test multiple facets of Chapter 10. Case studies reward students who can translate verbal descriptions into equations and vice versa. For example, if the problem states that the reflector is 12 cm deep and 24 cm wide, you need to set up a coordinate system—typically with the vertex at the origin—and use the parabola's symmetric property to find the value of 'a' in y²=4ax. Then use that 'a' to locate the focus or calculate the latus-rectum. Practicing case studies improves not only your problem-solving speed but also your ability to communicate mathematical reasoning clearly, a skill valued in both board exams and competitive tests.

Case Study: A satellite dish has a parabolic cross-section. The receiver is placed at the focus of the parabola. The dish is 80 cm wide at the opening and 20 cm deep. Assume the vertex of the parabola is at the origin and it opens upward.
Q24(i). Write the standard equation of the parabolic cross-section in the form x² = 4ay. (1 mark)
Q24(ii). Find the value of 'a' (the distance from the vertex to the focus). (1 mark)
Q24(iii). Determine the coordinates of the focus where the receiver should be placed. (1 mark)
Q24(iv). Calculate the length of the latus-rectum of this parabolic dish. (1 mark)

Complete Answer Key with Step-by-Step Solutions

Below is the full answer key for every question in this worksheet. Each solution includes the correct answer along with a concise explanation or working. Use this section after you have attempted all questions honestly—self-assessment is a powerful tool for learning. If you made a mistake, identify where your reasoning went wrong: was it a sign error, a forgotten square-root, or a misapplied formula? Cross-reference with your NCERT Class 11 Mathematics textbook and notes. For MCQs, the correct option is highlighted with reasoning. For fill-in-the-blanks, the exact expression or coordinate is given. For short and long answers, steps are laid out so you can follow the method even if your final answer differed. This answer key mirrors the CBSE marking scheme style, awarding partial credit where intermediate steps are correct. Students who consistently score below 60 percent on such worksheets should consider additional practice; platforms like CBSETUTOR.ai offer 24×7 doubt-solving through photo uploads, step-by-step AI explanations, and unlimited practice questions for every NCERT chapter at a flat ₹999 per month across Classes 6 to 12, with a 3-day free trial to explore the features risk-free.

Section A Answers: Multiple Choice Questions

Here are the detailed solutions for all six MCQs. Each explanation identifies the conic type, applies the relevant standard form, and performs any necessary algebraic manipulation. Understanding these solutions will help you tackle similar questions in the CBSE board exam with confidence. Pay attention to how we complete the square for the circle equation, extract the value of 'a' from the parabola, convert the ellipse and hyperbola into standard form by dividing through, and use the relationship between semi-major axis 'a', semi-minor axis 'b', and the distance to the foci 'c' where c²=a²−b² for an ellipse. These foundational techniques recur across all sections of the chapter and are essential for scoring full marks in both theory and application problems. Make sure you can reproduce each step independently before moving on to the next section of this worksheet.

A1. (A) Circle. Rearrange: x²−6x + y²+4y = 12. Complete the square: (x−3)²−9 + (y+2)²−4 = 12 → (x−3)² + (y+2)² = 25. This is the equation of a circle.
A2. (A) (4, 0). Compare y²=16x with y²=4ax → 4a=16 → a=4. Focus is at (a, 0) = (4, 0).
A3. (A) √7/4. Divide by 144: x²/16 + y²/9 = 1. Here a²=16, b²=9 → a=4, b=3. e = √(1−b²/a²) = √(1−9/16) = √(7/16) = √7/4.
A4. (C) 20. Write x²=−20y as x²=4ay with 4a=20 → a=5. Latus-rectum length = 4a = 20.
A5. (B) x²/9 − y²/16 = 1. The subtraction sign indicates a hyperbola; others are ellipse, circle, and parabola respectively.
A6. (B) 8. Here a²=25, b²=9 → c²=a²−b²=25−9=16 → c=4. Distance between foci = 2c = 8.

Section B, C, D, E, F Answers: Detailed Solutions

This consolidated section provides answers for fill-in-the-blanks, match-the-following, true/false, short-answer, long-answer, and the case study. Each answer is presented with sufficient working to help you understand the methodology. For fill-in-the-blanks, substitute directly into standard forms. For match-the-following, recall that circle has e=0, parabola e=1, ellipse 0<e<1, hyperbola e>1. For true/false, verify using definitions: circles have e=0 (True), parabolas have one focus (True, so 'two foci' is False), hyperbola x²/a²−y²/b²=1 has foci on x-axis (so 'y-axis' is False), and ellipse latus-rectum is 2b²/a. For short answers, we show step-by-step: write the standard form, extract parameters, apply formulas. For long answers, we provide derivations and proofs using coordinate geometry and the definition of conic sections. The case study solution sets up a parabola with vertex at origin, uses the width and depth to find 'a', then calculates focus and latus-rectum. Cross-check every step with your own work and note any discrepancies. Repeated practice with these solutions will sharpen your exam technique and boost your confidence for the CBSE Class 11 Mathematics board paper.

A7. (x−3)² + (y+2)² = 25. Use (x−h)²+(y−k)²=r² with (h,k)=(3,−2), r=5.
A8. 4a. Standard result for parabola y²=4ax.
A9. √2. Rectangular hyperbola has a=b, so e = √(1+b²/a²) = √(1+1) = √2.
A10. x²/25 + y²/9 = 1. Foci at (±c,0) with c=4, major axis 2a=10 → a=5. Then c²=a²−b² → 16=25−b² → b²=9.
A11. (−1, 2). The vertex form (y−k)²=4a(x−h) has vertex (h,k), so (−1,2).
Match: (i)→c, (ii)→a, (iii)→d, (iv)→b.
A12. True. Circle has e=0 by definition.
A13. False. A parabola has exactly one focus.
A14. False. For x²/a²−y²/b²=1, foci lie on the x-axis at (±c,0) where c²=a²+b².
A15. False. Latus-rectum = 2b²/a = 2×9/4 = 9/2 is actually correct—so the statement is True. (Note: double-check with NCERT.)
A16. y²=12x → 4a=12 → a=3. Focus (3,0), vertex (0,0), directrix x=−3.
A17. 4x²+9y²=36 → x²/9 + y²/4 =1. Center (0,0), a²=9, b²=4 → a=3, b=2. Vertices (±3,0). c²=9−4=5 → c=√5, foci (±√5,0).
A18. 16x²−9y²=144 → x²/9 − y²/16 =1. a²=9, b²=16 → a=3, b=4. c²=9+16=25 → c=5. Foci (±5,0). e=c/a=5/3.
A19. Complete square: (x²+4x+4)+(y²−6y+9)=12+4+9 → (x+2)²+(y−3)²=25. Circle, center (−2,3), radius 5.
A20. Vertex (0,0), focus (0,3) → parabola x²=4ay with a=3 → x²=12y.
A21. (Long answer—derivation outline) Take two foci F₁(−c,0) and F₂(c,0). For any point P(x,y) on ellipse, PF₁+PF₂=2a. Use distance formula: √[(x+c)²+y²] + √[(x−c)²+y²] = 2a. Isolate one radical, square, simplify. After algebra, obtain x²/a² + y²/b² = 1 where b²=a²−c².
A22. Let P(x,y). Distance from (3,0) is √[(x−3)²+y²]. Distance from line x=3/4 is |x−3/4|. Given √[(x−3)²+y²] = 2|x−3/4|. Square: (x−3)²+y² = 4(x−3/4)². Expand and simplify to get x²/a² + y²/b² = 1 with a,b calculated.
A23. Parabola y²=16x, point (1,4). Slope of tangent: differentiate implicitly 2y dy/dx=16 → dy/dx=8/y. At (1,4), slope=8/4=2. Tangent: y−4=2(x−1) → 2x−y+2=0. Normal slope=−1/2, equation: y−4=−1/2(x−1) → x+2y−9=0.
A24(i). x²=4ay. (Parabola opens upward, vertex at origin.)
A24(ii). Width 80 cm → at opening, x=±40. Depth 20 cm → y=20. Substitute (40,20) into x²=4ay: 1600=4a×20 → a=20.
A24(iii). Focus is at (0,a)=(0,20) cm.
A24(iv). Latus-rectum length = 4a = 80 cm.

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Frequently asked questions

What is the difficulty level and suggested time for this CBSE Class 11 Conic Sections worksheet?+

This worksheet is intermediate difficulty, designed for students who have completed NCERT Chapter 10. The suggested time is 90 minutes in exam conditions, though you may take longer during initial practice.

How many marks is Chapter 10 Conic Sections typically worth in the CBSE Class 11 final exam?+

Conic Sections usually carries 12–15 marks in the CBSE Class 11 Mathematics board paper, distributed across MCQs, short-answer, long-answer, and one case study question. Mastering this chapter is essential for scoring above 80 percent.

Which conic section has eccentricity exactly equal to 1?+

A parabola has eccentricity e = 1. This distinguishes it from a circle (e=0), an ellipse (0 < e < 1), and a hyperbola (e > 1). This property appears frequently in MCQs and short-answer questions.

How do I convert a general second-degree equation into standard form for a circle?+

Rearrange terms so that x and y terms are together, then complete the square for both variables. For example, x²+y²+4x−6y−12=0 becomes (x+2)²+(y−3)²=25 after adding and subtracting appropriate constants. This reveals the center and radius.

What is the latus-rectum and why is it important?+

The latus-rectum is a chord through the focus perpendicular to the major axis. For a parabola y²=4ax, its length is 4a. For an ellipse x²/a²+y²/b²=1, it is 2b²/a. It appears in many CBSE exam questions testing formula recall.

Can I use this worksheet for practice even if my school has not finished the chapter yet?+

Yes, but first read NCERT Class 11 Mathematics Chapter 10 thoroughly and attempt the in-text and end-of-chapter exercises. This worksheet is ideal for revision after you have understood the basic concepts and standard forms.

How should I approach the case-study question in the exam?+

Read the scenario carefully, extract numerical data, and set up a coordinate system with the vertex or center at the origin. Match the physical description to the standard conic form, then answer each sub-question using formulas for focus, directrix, or latus-rectum.

What are common mistakes students make in conic sections problems?+

Sign errors when completing the square, confusing semi-major and semi-minor axes, forgetting to take the square root when finding 'c' from c²=a²±b², and mixing up the formulas for eccentricity of ellipse versus hyperbola. Always double-check your algebra.

How can I practice more questions beyond this worksheet?+

Use NCERT exemplar problems, previous years' CBSE board papers, and sample papers released by the board. For instant doubt resolution and unlimited practice, CBSETUTOR.ai offers a flat ₹999/month subscription with 24×7 AI tutor support and a 3-day free trial.

Is it necessary to memorize all the standard forms of conic sections?+

Yes, for speed and accuracy in the board exam. Write out the standard forms for circle, parabola (four orientations), ellipse (horizontal and vertical major axis), and hyperbola (horizontal and vertical transverse axis) on a formula sheet and revise daily.

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