📐 CHAPTER 11: CONIC SECTIONS
Subject: Mathematics | Class: XI | Kendriya Vidyalaya Sangathan
📖 INTRODUCTION
A conic section is the curve obtained by intersecting a right circular cone with a plane. The four basic conic sections are:
🔹 Types of Conic Sections:
- Circle: When the plane is perpendicular to the axis of the cone
- Parabola: When the plane is parallel to a generator of the cone
- Ellipse: When the plane cuts the cone at an angle to the axis
- Hyperbola: When the plane cuts both nappes of the cone
🔹 Eccentricity (e)
- Circle: e = 0
- Parabola: e = 1
- Ellipse: 0 < e < 1
- Hyperbola: e > 1
⭕ CIRCLE
A circle is the set of all points in a plane that are at a fixed distance (radius) from a fixed point (centre).
🔹 Standard Equation of a Circle:
(x - h)² + (y - k)² = r²
where (h, k) is the centre and r is the radius.
🔹 Special Cases:
- Centre at origin: x² + y² = r²
- General form: x² + y² + 2gx + 2fy + c = 0
- Centre: (-g, -f)
- Radius: r = √(g² + f² - c)
🔄 PARABOLA
A parabola is the set of all points in a plane that are equidistant from a fixed line (directrix) and a fixed point (focus).
🔹 Standard Equations of Parabola:
| Equation | Vertex | Focus | Directrix | Axis | Latus Rectum |
|---|---|---|---|---|---|
| y² = 4ax | (0, 0) | (a, 0) | x = -a | x-axis | 4a |
| y² = -4ax | (0, 0) | (-a, 0) | x = a | x-axis | 4a |
| x² = 4ay | (0, 0) | (0, a) | y = -a | y-axis | 4a |
| x² = -4ay | (0, 0) | (0, -a) | y = a | y-axis | 4a |
🔹 Key Terms:
- Focus: Fixed point (a, 0) or (0, a)
- Directrix: Fixed line x = -a or y = -a
- Vertex: Point where axis meets the parabola
- Latus Rectum: Chord through focus perpendicular to axis, length = 4a
🟢 ELLIPSE
An ellipse is the set of all points in a plane such that the sum of distances from two fixed points (foci) is constant.
🔹 Standard Equations of Ellipse:
x²/a² + y²/b² = 1 (a > b, major axis along x-axis)
x²/b² + y²/a² = 1 (a > b, major axis along y-axis)
🔹 Key Terms:
- Major axis: 2a
- Minor axis: 2b
- Foci: (±ae, 0) or (0, ±ae)
- Eccentricity: e = √(1 - b²/a²)
- Latus Rectum: 2b²/a
- Vertices: (±a, 0) or (0, ±a)
- c² = a² - b² (where c = ae)
🔶 HYPERBOLA
A hyperbola is the set of all points in a plane such that the difference of distances from two fixed points (foci) is constant.
🔹 Standard Equations of Hyperbola:
x²/a² - y²/b² = 1 (transverse axis along x-axis)
y²/a² - x²/b² = 1 (transverse axis along y-axis)
🔹 Key Terms:
- Transverse axis: 2a
- Conjugate axis: 2b
- Foci: (±ae, 0) or (0, ±ae)
- Eccentricity: e = √(1 + b²/a²)
- Latus Rectum: 2b²/a
- Vertices: (±a, 0) or (0, ±a)
- c² = a² + b² (where c = ae)
📝 MCQ QUESTIONS
Q1. The equation of circle of radius 5 units touches the coordinate axes in the second quadrant is:
(a) x² + y² + 10x - 10y + 25 = 0
(b) x² + y² - 10x - 10y + 25 = 0
(c) x² + y² + 10x + 10y + 25 = 0
(d) x² + y² - 10x - 10y - 25 = 0
Answer: (a) x² + y² + 10x - 10y + 25 = 0
Solution: Centre = (-5, 5), r = 5
(x+5)² + (y-5)² = 25 ⇒ x² + y² + 10x - 10y + 25 = 0
Q2. Equation of the circle through the origin and cutting intercepts 2 and 4 on the coordinate axes is:
(a) x² + y² - 2x - 4y = 0
(b) x² + y² + 2x + 4y = 0
(c) x² + y² + 4x + 8y = 0
(d) x² + y² - 4x - 8y = 0
Answer: (a) x² + y² - 2x - 4y = 0
Solution: Circle passes through (0,0), (2,0), (0,4)
x² + y² + 2gx + 2fy = 0
(2,0): 4 + 4g = 0 ⇒ g = -1
(0,4): 16 + 8f = 0 ⇒ f = -2
x² + y² - 2x - 4y = 0
Q3. Equation (4m - 3)x² + my² + 10x - 8y + 16 = 0 represents a circle for m =
(a) 3 (b) 3/4 (c) 1 (d) -1
Answer: (c) 1
Solution: For circle, coefficient of x² = coefficient of y²
4m - 3 = m ⇒ 3m = 3 ⇒ m = 1
Q4. The radius of the circle x² + y² + 8x + 10y - 8 = 0 is:
(a) 8 (b) 10 (c) 9 (d) 7
Answer: (d) 7
Solution: r = √(16 + 25 + 8) = √49 = 7
Q5. The centre of the circle x² + y² + 8x + 10y - 8 = 0 is:
(a) (-4, -5) (b) (4, 5) (c) (-8, -10) (d) (8, 10)
Answer: (a) (-4, -5)
Q6. Find the equation of the circle with centre (-3, 2) and radius 4.
(a) (x - 3)² + (y - 2)² = 16
(b) (x + 3)² + (y + 2)² = 16
(c) (x - 3)² + (y + 2)² = 16
(d) (x + 3)² + (y - 2)² = 16
Answer: (d) (x + 3)² + (y - 2)² = 16
Q7. Eccentricity of circle is:
(a) 0 (b) 1 (c) 1.5 (d) 0.5
Answer: (a) 0
Q8. Eccentricity of parabola is:
(a) 0 (b) 1 (c) 1.5 (d) 0.5
Answer: (b) 1
Q9. Find the equation of the parabola with vertex at (0, 0) and focus at (0, 2).
(a) x² = 8y (b) x² = -8y (c) y² = 8x (d) y² = -8x
Answer: (a) x² = 8y
Solution: Focus (0, a) = (0, 2) ⇒ a = 2 ⇒ x² = 4ay = 8y
Q10. The directrix of parabola y² = 48x is:
(a) x = -12 (b) x = 12 (c) x = 24 (d) x = -24
Answer: (a) x = -12
Solution: y² = 4ax ⇒ 4a = 48 ⇒ a = 12
Directrix: x = -a = -12
Q11. What point of the parabola x² = 9y is the abscissa three times that of ordinate?
(a) (1, 1) (b) (3, 1) (c) (-3, -1) (d) (-3, -3)
Answer: (b) (3, 1)
Solution: Let point be (3y, y). x² = 9y ⇒ (3y)² = 9y ⇒ 9y² = 9y ⇒ y = 1
Point = (3, 1)
Q12. The point where the axis meets the parabola is called:
(a) Latus rectum (b) Tangent (c) Chord (d) Vertex
Answer: (d) Vertex
Q13. The vertex of the parabola (y + a)² = 8a(x - a) is:
(a) (-a, -a) (b) (a, a) (c) (-a, a) (d) (a, -a)
Answer: (d) (a, -a)
Solution: Put Y = y + a, X = x - a ⇒ Y² = 8aX
Vertex: X = 0, Y = 0 ⇒ x - a = 0, y + a = 0 ⇒ (a, -a)
Q14. The length of latus rectum of parabola y² = 16x is:
(a) -16 (b) -4 (c) 16 (d) 4
Answer: (c) 16
Solution: 4a = 16 ⇒ a = 4, Latus Rectum = 4a = 16
Q15. The focus of parabola x² = 16y is:
(a) (-4, 0) (b) (4, 0) (c) (0, 4) (d) (0, -4)
Answer: (c) (0, 4)
Solution: x² = 4ay ⇒ 4a = 16 ⇒ a = 4, Focus = (0, a) = (0, 4)
Q16. In an ellipse, the distance between its foci is 6 and its minor axis is 8, then its eccentricity is:
(a) 4/5 (b) 1/√52 (c) 3/5 (d) 1/2
Answer: (c) 3/5
Solution: 2c = 6 ⇒ c = 3, 2b = 8 ⇒ b = 4
a² = b² + c² = 16 + 9 = 25 ⇒ a = 5
e = c/a = 3/5
Q17. The difference between the length of the major axis and latus rectum of an ellipse is:
(a) 2ae² (b) ae² (c) 2ae (d) ae
Answer: (a) 2ae²
Solution: Major axis = 2a, Latus Rectum = 2b²/a
Difference = 2a - 2b²/a = 2(a² - b²)/a = 2c²/a = 2ae²
Q18. The eccentricity of ellipse, if the minor axis is equal to the distance between the foci, is:
(a) √3/2 (b) 2/√3 (c) √2/3 (d) 1/√2
Answer: (d) 1/√2
Solution: 2b = 2c ⇒ b = c
a² = b² + c² = 2c² ⇒ a = √2c
e = c/a = 1/√2
Q19. The eccentricity of ellipse, 5x² + 9y² = 1, is:
(a) 3/4 (b) 4/5 (c) 2/3 (d) 1/2
Answer: (c) 2/3
Solution: x²/(1/5) + y²/(1/9) = 1 ⇒ a² = 1/5, b² = 1/9
e = √(1 - b²/a²) = √(1 - 5/9) = √(4/9) = 2/3
Q20. The eccentricity of rectangular hyperbola is:
(a) 1 (b) √2 (c) √3 (d) 2
Answer: (b) √2
Solution: For rectangular hyperbola, a = b ⇒ e = √(1 + b²/a²) = √2
📚 CASE BASED QUESTIONS
Q21. Case Study: "C: x² + y² - 2x - 2ay - 8 = 0, a is variable." Gaurang is analyzing the equation of the given curve and the dependency of curve with 'a'.
(i) The equation represents the family of circles having center in first quadrant, if 'a' is:
(a) Any Real number (b) 0 (c) Always a negative real number (d) Always a positive real number
Answer: (d) Always a positive real number
Solution: Centre = (1, a), for first quadrant a > 0
(ii) If a = 0, then center of C is:
(a) (1, 0) (b) (0, 1) (c) (8, 0) (d) (0, 8)
Answer: (a) (1, 0)
(iii) If a = 0, then radius of C is:
(a) 1 (b) 3 (c) 2 (d) 4
Answer: (b) 3
Solution: x² + y² - 2x - 8 = 0 ⇒ (x-1)² + y² = 9 ⇒ r = 3
(iv) Curve C passes through 2 fixed points whose coordinates are:
(a) (-2, 0), (4, 0) (b) (2, 0), (4, 0) (c) (-4, 0), (4, 0) (d) (2, 0), (-4, 0)
Answer: (a) (-2, 0), (4, 0)
Solution: Put y = 0: x² - 2x - 8 = 0 ⇒ (x-4)(x+2) = 0
(v) Eccentricity of the Curve C is:
(a) Depend on the value of a (b) Always 0 (c) Always 1 (d) Always less than 1
Answer: (b) Always 0
Reason: C is a circle, eccentricity = 0
Q22. The focus of a parabolic mirror is at a distance of 5 cm from its vertex and the mirror is 45 cm deep.
(i) What is the equation of the parabolic section?
(a) y² = -5x (b) y² = 45x (c) y² = 5x (d) y² = 20x
Answer: (d) y² = 20x
Solution: Focus at (5, 0) ⇒ a = 5 ⇒ y² = 4ax = 20x
(ii) What is the length of distance AB?
(a) 5 cm (b) 45 cm (c) 30 cm (d) 20 cm
Answer: (c) 30 cm
(iii) What is the coordinate of focus point of the parabolic mirror?
(a) (5, 0) (b) (20, 0) (c) (45, 30) (d) (45, 15)
Answer: (a) (5, 0)
(iv) What is the coordinate of point A of the parabolic mirror?
(a) (5, 0) (b) (45, 5) (c) (45, 20) (d) (45, 15)
Answer: (c) (45, 20)
(v) What is the equation of axis of the parabolic mirror?
(a) x = 0 (b) y = 0 (c) x = 45 (d) y = 45
Answer: (b) y = 0
Q23. Based on the ellipse information:
(i) If equation of ellipse is (x - 2)² + 9(y - 1)² = 9, then length of major axis is:
(a) 3 (b) 6 (c) 9 (d) 18
Answer: (b) 6
Solution: (x-2)²/9 + (y-1)²/1 = 1 ⇒ a = 3 ⇒ Major axis = 2a = 6
(ii) If equation of ellipse is (x - 2)² + 9(y - 1)² = 9, then centre is:
(a) (2, 1) (b) (3, 1) (c) (-2, 1) (d) (1, 9)
Answer: (a) (2, 1)
(iii) If equation of ellipse is 9(x - 2)² + (y - 1)² = 9, then major axis is parallel to:
(a) x + y = 0 (b) y = 0 (c) x + y = 0 (d) x = 0
Answer: (d) x = 0
Solution: (x-2)²/1 + (y-1)²/9 = 1 ⇒ major axis along y-axis
(iv) If equation of ellipse is (x - 2)² + 25(y - 1)² = 25, then coordinates of vertices are:
(a) (2 ± 5, 1) (b) (2, 1 ± 1) (c) (2 ± 5, 0) (d) (±5, 0)
Answer: (a) (2 ± 5, 1)
(v) If equation of ellipse is (x - 2)² + 9(y - 1)² = 9, then foci are:
(a) (2 ± 2√2, 1) (b) (2, 1 ± 2√2) (c) (2, 1) (d) (±2√2, 0)
Answer: (a) (2 ± 2√2, 1)
Solution: a² = 9, b² = 1 ⇒ c² = a² - b² = 8 ⇒ c = 2√2
🧠 ASSERTION & REASON QUESTIONS
Q26. Assertion (A): The conic section having the equation 2x² - 2y² - 5 = 0 is a circle.
Reason (R): The conic section having general equation Ax² + Cy² + Dx + Ey + F = 0 is a circle, if A = C.
Answer: (d) A is false but R is true.
Explanation: 2x² - 2y² = 5 ⇒ x² - y² = 5/2 is a hyperbola, not a circle.
Q27. Assertion (A): The point (1.5, 2) is inside the circle S: x² + y² - 5x - 4y = 0.
Reason (R): The point (x₁, y₁) is inside, on or outside the circle S₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c, according as S₁ < 0; or = 0; or > 0.
Answer: (a) Both A and R are true and R is the correct explanation of A.
Solution: S = 2.25 + 4 - 7.5 - 8 = -9.25 < 0 ⇒ inside
Q28. Assertion (A): The parabola x² + 6y = 0 opens upwards along y-axis.
Reason (R): When the axis of symmetry is along the X-axis, the parabola opens upwards.
Answer: (c) A is true but R is false.
Explanation: x² = -6y ⇒ Opens downwards along y-axis. R is false.
Q29. Assertion (A): The major axis of the ellipse 5x² + 9y² = 1 is along x-axis.
Reason (R): The major axis is along the x-axis if the coefficient of x² has the larger denominator.
Answer: (a) Both A and R are true and R is the correct explanation of A.
Solution: x²/(1/5) + y²/(1/9) = 1, 1/5 > 1/9 ⇒ major axis along x-axis
Q30. Assertion (A): Eccentricity of the hyperbola x² - y² = 1 is always less than 1.
Reason (R): For the hyperbola, a² = c² - b², and Eccentricity = c/a, so the eccentricity is always less than one.
Answer: (d) A is false but R is true.
Explanation: For hyperbola, e > 1 always. x² - y² = 1 ⇒ a = 1, b = 1 ⇒ e = √(1+1) = √2 > 1
📝 VERY SHORT ANSWER TYPE QUESTIONS
Q31. Find equation of circle with centre (-a, -a) and radius √2a.
Answer: (x + a)² + (y + a)² = 2a²
⇒ x² + y² + 2ax + 2ay = 0
Q32. Find equation of circle whose end points of diameter are (a, 0) and (0, b).
Answer: x² + y² - ax - by = 0
Q33. Find radius and centre of circle: 3x² + 3y² + 9x - 6y + 1 = 0.
Answer: Divide by 3: x² + y² + 3x - 2y + 1/3 = 0
Centre = (-3/2, 1)
r = √(9/4 + 1 - 1/3) = √(9/4 + 2/3) = √(35/12)
Q34. Find equation of circle with centre (1, 5) and passing through the point (7, -1).
Answer: r² = (7-1)² + (-1-5)² = 36 + 36 = 72
(x - 1)² + (y - 5)² = 72
Q35. Does point (7, -5) lie inside, outside or on the circle x² + y² - 6x + 8y + 9 = 0?
Answer: S = 49 + 25 - 42 - 40 + 9 = 1 > 0 ⇒ Outside
Q36. For what value of k, does the equation 9x² + y² = k(x² - y² - 2x) represent the equation of a circle?
Answer: 9x² + y² = kx² - ky² - 2kx
(9-k)x² + (1+k)y² + 2kx = 0
For circle, coefficient of x² = coefficient of y²
9 - k = 1 + k ⇒ 8 = 2k ⇒ k = 4
Q37. Find equation of the parabola with vertex at origin and directrix y - 2 = 0.
Answer: Directrix: y = 2 ⇒ a = -2
x² = 4ay ⇒ x² = -8y
Q38. Find equation of the parabola with vertex at origin, axis along x-axis and passing through the point (3, 2).
Answer: y² = 4ax, (3,2): 4 = 12a ⇒ a = 1/3
y² = 4x/3
Q39. Find equation of the parabola with focus (3, -4) and directrix x + 1 = 0.
Answer: By definition: √[(x-3)² + (y+4)²] = |x+1|
(x-3)² + (y+4)² = (x+1)²
x² - 6x + 9 + y² + 8y + 16 = x² + 2x + 1
y² + 8y + 25 - 8x - 1 = 0 ⇒ y² + 8y - 8x + 24 = 0
Q40. Find focus, vertex, and directrix of the parabola y² = x.
Answer: y² = 4ax ⇒ 4a = 1 ⇒ a = 1/4
Vertex = (0, 0)
Focus = (1/4, 0)
Directrix: x = -1/4
Q41. Determine the equation of the ellipse whose foci are (4, 0) and (-4, 0), eccentricity = 1/2.
Answer: c = 4, e = 1/2 ⇒ a = c/e = 8
b² = a² - c² = 64 - 16 = 48
x²/64 + y²/48 = 1
Q42. If the distance between the foci of a hyperbola is 16 and eccentricity is √2, then obtain its equation.
Answer: 2c = 16 ⇒ c = 8, e = √2
a = c/e = 8/√2 = 4√2
b² = c² - a² = 64 - 32 = 32
x²/32 - y²/32 = 1
Q43. In an ellipse, the distance between its foci is 10 and its minor axis is 8, then its eccentricity.
Answer: 2c = 10 ⇒ c = 5, 2b = 8 ⇒ b = 4
a² = b² + c² = 16 + 25 = 41 ⇒ a = √41
e = c/a = 5/√41
Q44. Find the equation of diameter of a circle: x² + y² + 2x - 4y = 4, that is parallel to 3x + 5y = 4.
Answer: Centre = (-1, 2)
Diameter is line through centre parallel to 3x + 5y = 4
3x + 5y = 3(-1) + 5(2) = 7
3x + 5y = 7
Q45. Find equation of a parabola whose vertex is at origin and which is symmetric about the x-axis and passes through the point (-2, -3).
Answer: y² = 4ax, (-2,-3): 9 = -8a ⇒ a = -9/8
y² = -9x/2
✅ ANSWERS - MCQs
| Q.No. | Answer | Q.No. | Answer |
|---|---|---|---|
| 1 | (a) | 11 | (b) |
| 2 | (a) | 12 | (d) |
| 3 | (c) | 13 | (d) |
| 4 | (d) | 14 | (c) |
| 5 | (a) | 15 | (c) |
| 6 | (d) | 16 | (c) |
| 7 | (a) | 17 | (a) |
| 8 | (b) | 18 | (d) |
| 9 | (a) | 19 | (c) |
| 10 | (a) | 20 | (b) |
📌 KEY FORMULAS SUMMARY
| Conic | Equation | Eccentricity | Key Terms |
|---|---|---|---|
| Circle | x² + y² = r² | e = 0 | Centre: (0,0), Radius: r |
| Parabola | y² = 4ax | e = 1 | Focus: (a,0), Directrix: x = -a |
| Ellipse | x²/a² + y²/b² = 1 | 0 < e < 1 | a > b, c² = a² - b² |
| Hyperbola | x²/a² - y²/b² = 1 | e > 1 | c² = a² + b² |
🔹 Important Formulas:
- Circle: (x-h)² + (y-k)² = r²
- Parabola: y² = 4ax, Latus Rectum = 4a
- Ellipse: e = √(1 - b²/a²), Latus Rectum = 2b²/a
- Hyperbola: e = √(1 + b²/a²), Latus Rectum = 2b²/a
📚 Kendriya Vidyalaya Sangathan | Class XI Mathematics
Chapter 11: Conic Sections | Student Support Material