📐 Chapter 8: Conic Section | NEB Mathematics Notes Class 12
Based on Latest Syllabus 2080 | Complete Notes for Circle, Parabola, Ellipse, Hyperbola
✅ Updated according to latest syllabus of 2080 | All exercises included
🔺 Conic Section — This chapter covers the study of curves obtained by intersecting a cone with a plane: Circle, Parabola, Ellipse, and Hyperbola. The notes have been updated according to the latest syllabus of 2080. Now you don’t need to go anywhere searching for the notes of this chapter because we are here to serve you.
📚 Exercises
📖 Introduction to Conic Section
Right Cone: Let O be the fixed point and OC the fixed line. The surface generated by rotating the line OA around OC such that ∠AOC is always constant is known as the right cone.
• Vertex: Point O
• Axis: Line OC
• Generator: Line OA
• Semi-vertical angle: ∠AOC
• Vertex: Point O
• Axis: Line OC
• Generator: Line OA
• Semi-vertical angle: ∠AOC
Double Right Cone: If the cone OA’B’ is symmetrical to the cone OAB about OC’ opposite to OC, then ABOA’B’ is said to be the double right cone.
📐 Types of Conic Sections
① Circle
If a plane intersects a cone perpendicular to the axis, then the section is a circle.
② Ellipse
If a plane intersects a cone at a given angle with the axis greater than the semi-vertical angle, then the section is an ellipse.
③ Parabola
If an intersecting plane, not passing through the vertex, is parallel to the generator of the cone, then the section is a parabola.
④ Hyperbola
If a plane intersects the double right cone such that the angle between the axis and the plane is less than the semi-vertical angle, then the section is a hyperbola.
🎯 General Definition of Conic Section
A conic section is the locus of a point which moves in a plane in such a way that the ratio of its distance from a fixed point (focus) to its distance from a fixed line (directrix) is always constant.
\[ \frac{\text{Distance from focus}}{\text{Distance from directrix}} = e \quad \text{(eccentricity)} \]
• Focus (S): Fixed point
• Directrix: Fixed straight line
• Eccentricity (e): Constant ratio
• Axis: Straight line passing through focus and perpendicular to directrix
• Vertex: Intersection of the curve and the axis
\[ \frac{\text{Distance from focus}}{\text{Distance from directrix}} = e \quad \text{(eccentricity)} \]
• Focus (S): Fixed point
• Directrix: Fixed straight line
• Eccentricity (e): Constant ratio
• Axis: Straight line passing through focus and perpendicular to directrix
• Vertex: Intersection of the curve and the axis
● Circle
Definition: The locus of a moving point such that its distance from a fixed point is always constant is called a circle.
• Fixed point = Centre
• Constant distance = Radius
Standard Equation: \(x^2 + y^2 = a^2\) (centre at origin, radius a)
General Equation: \(x^2 + y^2 + 2gx + 2fy + c = 0\) (centre at \((-g, -f)\), radius = \(\sqrt{g^2 + f^2 – c}\))
• Fixed point = Centre
• Constant distance = Radius
Standard Equation: \(x^2 + y^2 = a^2\) (centre at origin, radius a)
General Equation: \(x^2 + y^2 + 2gx + 2fy + c = 0\) (centre at \((-g, -f)\), radius = \(\sqrt{g^2 + f^2 – c}\))
➰ Parabola (\(e = 1\))
Standard Equations:
• \(y^2 = 4ax\) (opens right), Vertex: (0,0), Focus: (a,0), Directrix: \(x = -a\)
• \(y^2 = -4ax\) (opens left)
• \(x^2 = 4ay\) (opens up)
• \(x^2 = -4ay\) (opens down)
Latus Rectum: Length = \(4a\)
• \(y^2 = 4ax\) (opens right), Vertex: (0,0), Focus: (a,0), Directrix: \(x = -a\)
• \(y^2 = -4ax\) (opens left)
• \(x^2 = 4ay\) (opens up)
• \(x^2 = -4ay\) (opens down)
Latus Rectum: Length = \(4a\)
⬭ Ellipse (\(0 < e < 1\))
Standard Equation (Horizontal Major Axis):
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad \text{where } b^2 = a^2(1 – e^2)
\]
• Centre: (0,0)
• Vertices: \((\pm a, 0)\)
• Foci: \((\pm ae, 0)\)
• Directrices: \(x = \pm \frac{a}{e}\)
• Latus Rectum: \(\frac{2b^2}{a}\)
Standard Equation (Vertical Major Axis): \[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1, \quad a > b \]
• Vertices: \((\pm a, 0)\)
• Foci: \((\pm ae, 0)\)
• Directrices: \(x = \pm \frac{a}{e}\)
• Latus Rectum: \(\frac{2b^2}{a}\)
Standard Equation (Vertical Major Axis): \[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1, \quad a > b \]
⤨ Hyperbola (\(e > 1\))
Standard Equation (Horizontal Transverse Axis):
\[
\frac{x^2}{a^2} – \frac{y^2}{b^2} = 1, \quad \text{where } b^2 = a^2(e^2 – 1)
\]
• Centre: (0,0)
• Vertices: \((\pm a, 0)\)
• Foci: \((\pm ae, 0)\)
• Directrices: \(x = \pm \frac{a}{e}\)
• Latus Rectum: \(\frac{2b^2}{a}\)
• Asymptotes: \(y = \pm \frac{b}{a}x\)
Standard Equation (Vertical Transverse Axis): \[ \frac{y^2}{a^2} – \frac{x^2}{b^2} = 1 \]
• Vertices: \((\pm a, 0)\)
• Foci: \((\pm ae, 0)\)
• Directrices: \(x = \pm \frac{a}{e}\)
• Latus Rectum: \(\frac{2b^2}{a}\)
• Asymptotes: \(y = \pm \frac{b}{a}x\)
Standard Equation (Vertical Transverse Axis): \[ \frac{y^2}{a^2} – \frac{x^2}{b^2} = 1 \]
📏 A Line and a Circle
Let \(y = mx + c\) and \(x^2 + y^2 = a^2\) be the equations of a line and a circle respectively.
To find the points of intersection, substitute \(y = mx + c\) into the circle equation:
\[ x^2 + (mx + c)^2 = a^2 \] \[ x^2 + m^2x^2 + 2mcx + c^2 – a^2 = 0 \] \[ (1 + m^2)x^2 + 2mcx + (c^2 – a^2) = 0 \] This quadratic gives the x-coordinates of intersection points. The line is:
To find the points of intersection, substitute \(y = mx + c\) into the circle equation:
\[ x^2 + (mx + c)^2 = a^2 \] \[ x^2 + m^2x^2 + 2mcx + c^2 – a^2 = 0 \] \[ (1 + m^2)x^2 + 2mcx + (c^2 – a^2) = 0 \] This quadratic gives the x-coordinates of intersection points. The line is:
- Tangent if discriminant = 0
- Secant if discriminant > 0
- Does not intersect if discriminant < 0
📊 Eccentricity Summary
e = 0 Circle
Special case of ellipse with e = 0
e = 1 Parabola
The conic is a parabola
0 < e < 1 Ellipse
The conic is an ellipse (circle when e = 0)
e > 1 Hyperbola
The conic is a hyperbola
📝 Exercise 8.1 – Sample Solutions
Q1 Find the equation of the circle with centre (0,0) and radius 5.
Solution: Equation of circle: \(x^2 + y^2 = r^2\)
Here \(r = 5\) ⇒ \(x^2 + y^2 = 25\)
Here \(r = 5\) ⇒ \(x^2 + y^2 = 25\)
✅ \(x^2 + y^2 = 25\)
Q2 Find the centre and radius of the circle \(x^2 + y^2 – 4x + 6y – 3 = 0\).
Solution: Complete the square:
\(x^2 – 4x + y^2 + 6y = 3\)
\((x^2 – 4x + 4) + (y^2 + 6y + 9) = 3 + 4 + 9 = 16\)
\((x – 2)^2 + (y + 3)^2 = 16\)
Centre = \((2, -3)\), Radius = \(4\)
\(x^2 – 4x + y^2 + 6y = 3\)
\((x^2 – 4x + 4) + (y^2 + 6y + 9) = 3 + 4 + 9 = 16\)
\((x – 2)^2 + (y + 3)^2 = 16\)
Centre = \((2, -3)\), Radius = \(4\)
✅ Centre \((2, -3)\), Radius \(4\)
Q3 Find the equation of the parabola with focus (2,0) and directrix \(x = -2\).
Solution: For parabola, \(e = 1\).
Focus at (a,0) with a = 2, directrix \(x = -a\).
Equation: \(y^2 = 4ax = 8x\)
Focus at (a,0) with a = 2, directrix \(x = -a\).
Equation: \(y^2 = 4ax = 8x\)
✅ \(y^2 = 8x\)
Q4 Find the equation of the ellipse with foci at \((\pm 3, 0)\) and eccentricity \(e = \frac{3}{5}\).
Solution: For ellipse, \(ae = 3\) ⇒ \(a \cdot \frac{3}{5} = 3\) ⇒ \(a = 5\)
\(b^2 = a^2(1 – e^2) = 25(1 – \frac{9}{25}) = 25 \times \frac{16}{25} = 16\) ⇒ \(b = 4\)
Equation: \(\frac{x^2}{25} + \frac{y^2}{16} = 1\)
\(b^2 = a^2(1 – e^2) = 25(1 – \frac{9}{25}) = 25 \times \frac{16}{25} = 16\) ⇒ \(b = 4\)
Equation: \(\frac{x^2}{25} + \frac{y^2}{16} = 1\)
✅ \(\frac{x^2}{25} + \frac{y^2}{16} = 1\)
Q5 Find the equation of the hyperbola with foci at \((\pm 5, 0)\) and eccentricity \(e = \frac{5}{3}\).
Solution: For hyperbola, \(ae = 5\) ⇒ \(a \cdot \frac{5}{3} = 5\) ⇒ \(a = 3\)
\(b^2 = a^2(e^2 – 1) = 9(\frac{25}{9} – 1) = 9 \times \frac{16}{9} = 16\) ⇒ \(b = 4\)
Equation: \(\frac{x^2}{9} – \frac{y^2}{16} = 1\)
\(b^2 = a^2(e^2 – 1) = 9(\frac{25}{9} – 1) = 9 \times \frac{16}{9} = 16\) ⇒ \(b = 4\)
Equation: \(\frac{x^2}{9} – \frac{y^2}{16} = 1\)
✅ \(\frac{x^2}{9} – \frac{y^2}{16} = 1\)
Q6 Find the condition for the line \(y = mx + c\) to be tangent to the circle \(x^2 + y^2 = a^2\).
Solution: Substituting \(y = mx + c\) into \(x^2 + y^2 = a^2\):
\(x^2 + (mx + c)^2 = a^2\) ⇒ \((1 + m^2)x^2 + 2mcx + (c^2 – a^2) = 0\)
For tangency, discriminant = \((2mc)^2 – 4(1 + m^2)(c^2 – a^2) = 0\)
\(4m^2c^2 – 4(1 + m^2)(c^2 – a^2) = 0\)
Dividing by 4: \(m^2c^2 – (1 + m^2)(c^2 – a^2) = 0\)
\(m^2c^2 – c^2 + a^2 – m^2c^2 + m^2a^2 = 0\) ⇒ \(-c^2 + a^2 + m^2a^2 = 0\)
\(a^2(1 + m^2) = c^2\) ⇒ \(c = \pm a\sqrt{1 + m^2}\)
\(x^2 + (mx + c)^2 = a^2\) ⇒ \((1 + m^2)x^2 + 2mcx + (c^2 – a^2) = 0\)
For tangency, discriminant = \((2mc)^2 – 4(1 + m^2)(c^2 – a^2) = 0\)
\(4m^2c^2 – 4(1 + m^2)(c^2 – a^2) = 0\)
Dividing by 4: \(m^2c^2 – (1 + m^2)(c^2 – a^2) = 0\)
\(m^2c^2 – c^2 + a^2 – m^2c^2 + m^2a^2 = 0\) ⇒ \(-c^2 + a^2 + m^2a^2 = 0\)
\(a^2(1 + m^2) = c^2\) ⇒ \(c = \pm a\sqrt{1 + m^2}\)
✅ Condition: \(c = \pm a\sqrt{1 + m^2}\)
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This PDF provides the solutions of every question from the 1st exercise of class 12 conic section. If you want the solutions of other exercises then you can select the exercise from the button given above.
This PDF provides the solutions of every question from the 1st exercise of class 12 conic section. If you want the solutions of other exercises then you can select the exercise from the button given above.
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📌 Key Formulas Summary:
• Circle: \(x^2 + y^2 = a^2\)
• Parabola: \(y^2 = 4ax\) (Focus: (a,0), Directrix: x = -a)
• Ellipse: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), \(b^2 = a^2(1 – e^2)\)
• Hyperbola: \(\frac{x^2}{a^2} – \frac{y^2}{b^2} = 1\), \(b^2 = a^2(e^2 – 1)\)
• Eccentricity: \(e = \frac{\text{Distance from focus}}{\text{Distance from directrix}}\)
• Circle: \(x^2 + y^2 = a^2\)
• Parabola: \(y^2 = 4ax\) (Focus: (a,0), Directrix: x = -a)
• Ellipse: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), \(b^2 = a^2(1 – e^2)\)
• Hyperbola: \(\frac{x^2}{a^2} – \frac{y^2}{b^2} = 1\), \(b^2 = a^2(e^2 – 1)\)
• Eccentricity: \(e = \frac{\text{Distance from focus}}{\text{Distance from directrix}}\)