Class 12 Mathematics CONIC SECTION Notes

Class 12 Mathematics CONIC SECTION Notes
Class 12 Mathematics Conic Section Notes | Nepal eNotes

Unit 10

Analytical Geometry

Class 12 Mathematics

Conic Section

Class 12 Mathematics – Conic Section Notes PDF

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Contents / Quick Navigation

1. Introduction to Conics 2. Circle: Basic Form 3. Condition of Tangency to Circle 4. Tangent & Normal to Circle 5. Standard Parabola 6. Tangent to Parabola 7. Normal to Parabola 8. Standard Ellipse 9. Standard Hyperbola 10. Conic Comparison 11. Worked Examples 12. Common Mistakes 13. Important Exam Questions 14. One-Minute Revision 15. Diagram Practice

NEB / CDC Focus

This unit covers the condition of tangency of a line to a circle, equations of tangent and normal to a circle at a given point, the standard equation of a parabola and its tangent and normal, and the standard equations of ellipse and hyperbola.

1. Introduction to Conic Sections

Conic Section

A conic section is a curve obtained by intersecting a plane with a double cone. The familiar non-degenerate conics are the circle, parabola, ellipse and hyperbola.

A useful unified description uses a fixed point called the focus, a fixed line called the directrix, and the ratio

e = distance from point to focus / perpendicular distance from point to directrix.

The constant e is called the eccentricity. For orientation:

ConicEccentricityTypical geometric character
Circlee=0All points are at a constant distance from the centre.
Ellipse0<e<1Closed oval curve.
Parabolae=1One open branch.
Hyperbolae>1Two separated open branches.
The Four Main Conics Circle Ellipse Parabola Hyperbola
Figure 1: Basic visual forms of the four standard conics.

2. Circle: Standard and General Forms

Circle

A circle is the locus of a point whose distance from a fixed point, the centre, is constant.

2.1 Standard Form

(x−h)²+(y−k)²=r²

Centre: (h,k), radius: r.

For a circle centred at the origin:

x²+y²=a².

2.2 General Form

x²+y²+2gx+2fy+c=0. Centre = (−g,−f) Radius = √(g²+f²−c)
Circle, Radius and Tangent C P radius tangent At the point of contact, the radius is perpendicular to the tangent.
Figure 2: The tangent to a circle is perpendicular to the radius at the point of contact.

3. Condition of Tangency of a Line to a Circle

A line is tangent to a circle if it meets the circle at exactly one point. Geometrically, the perpendicular distance from the centre to the line must equal the radius.

3.1 Circle x²+y²=a² and Line y=mx+c

The distance of the origin from mx−y+c=0 is

|c|/√(m²+1).

For tangency this equals a, hence

Tangency Condition
c²=a²(1+m²).

Therefore a line of slope m tangent to x²+y²=a² has form

y=mx±a√(1+m²).

3.2 General Circle and General Line

For

x²+y²+2gx+2fy+c=0

with centre (−g,−f) and radius r=√(g²+f²−c), the line

lx+my+n=0

is tangent if

|n−lg−mf| / √(l²+m²) = √(g²+f²−c).
Tangency Test: Distance = Radius perpendicular distance = r line
Figure 3: A line is tangent exactly when its perpendicular distance from the centre equals the radius.
Worked Example

Find the values of c for which y=2x+c is tangent to x²+y²=9.

c²=9(1+2²)=45.

c=±3√5.

4. Tangent and Normal to a Circle at a Given Point

4.1 Circle x²+y²=a²

If P(x₁,y₁) lies on the circle, then

Tangent at (x₁,y₁)
xx₁+yy₁=a².

The normal to a circle passes through the centre and the point of contact. For the circle centred at the origin, the normal is the line through (0,0) and (x₁,y₁).

Normal at (x₁,y₁)
x₁y−y₁x=0.

4.2 General Circle

For

x²+y²+2gx+2fy+c=0,

the tangent at (x₁,y₁) is

xx₁+yy₁+g(x+x₁)+f(y+y₁)+c=0.

The normal is the straight line joining the centre (−g,−f) to (x₁,y₁).

Worked Example

Find tangent and normal to x²+y²=25 at (3,4).

Tangent:

3x+4y=25.

Normal: the line through (0,0) and (3,4):

4x−3y=0.

5. Standard Equation of a Parabola

Parabola

A parabola is the locus of a point whose distance from a fixed point (focus) equals its perpendicular distance from a fixed line (directrix).

5.1 Parabola y²=4ax

y²=4ax Vertex: (0,0) Focus: (a,0) Directrix: x=−a Axis: y=0 Length of latus rectum: 4a

The standard parametric point is (at²,2at).

5.2 Other Standard Orientations

EquationFocusDirectrixOpening
y²=4ax(a,0)x=−aRight
y²=−4ax(−a,0)x=aLeft
x²=4ay(0,a)y=−aUp
x²=−4ay(0,−a)y=aDown
Parabola y² = 4ax F(a,0) V x=−a latus rectum
Figure 4: Main geometric parts of y²=4ax.

6. Tangent to a Parabola

6.1 Tangent at a Point

For y²=4ax, the tangent at (x₁,y₁) on the parabola is

yy₁=2a(x+x₁).

6.2 Parametric Tangent

At the point (at²,2at), the tangent is

ty=x+at².

6.3 Tangent in Slope Form

A tangent of slope m to y²=4ax is

y=mx+a/m,   m≠0.
Tangent to y²=4ax at P P(x₁,y₁) tangent
Figure 5: The tangent touches the parabola at the specified point P.
Worked Example

Find the tangent to y²=8x at (2,4).

Here 4a=8⇒a=2.

yy₁=2a(x+x₁) ⇒4y=4(x+2).

y=x+2.

7. Normal to a Parabola

The normal is perpendicular to the tangent at the point of contact. For y²=4ax, at (x₁,y₁) with y₁≠0, the tangent slope is 2a/y₁, so the normal slope is −y₁/(2a).

Point Form of Normal
y−y₁ = −(y₁/2a)(x−x₁).

7.1 Parametric Normal

At (at²,2at),

y = −tx + 2at + at³.
Tangent and Normal at the Same Point P tangent normal
Figure 6: At P, tangent and normal are perpendicular.
Worked Example

Find the normal to y²=8x at (2,4).

Here a=2, so normal slope =−4/(4)=−1.

y−4=−(x−2).

x+y−6=0.

8. Standard Equation of an Ellipse

Ellipse

An ellipse is a closed conic. In the standard horizontal form, the major axis lies along the x-axis.

8.1 Major Axis Along x-axis

x²/a² + y²/b² = 1,   a>b>0 c²=a²−b² Foci: (±c,0) Vertices: (±a,0) Eccentricity: e=c/a,   0<e<1

8.2 Major Axis Along y-axis

x²/b² + y²/a² = 1,   a>b>0 Foci: (0,±c),   c²=a²−b²
Ellipse x²/a² + y²/b² = 1 F₁F₂ −aa The foci lie inside the ellipse because c<a.
Figure 7: Standard horizontal ellipse and its foci.
Worked Example

For x²/25+y²/9=1, find the foci and eccentricity.

a²=25, b²=9 ⇒ c²=16 ⇒ c=4.

Foci: (±4,0).

e=c/a=4/5.

9. Standard Equation of a Hyperbola

Hyperbola

A hyperbola is a conic with two separate branches. In standard horizontal form, its transverse axis lies along the x-axis.

9.1 Transverse Axis Along x-axis

x²/a² − y²/b² = 1 c²=a²+b² Foci: (±c,0) Vertices: (±a,0) Eccentricity: e=c/a>1 Asymptotes: y=±(b/a)x

9.2 Transverse Axis Along y-axis

y²/a² − x²/b² = 1 Foci: (0,±c),   c²=a²+b² Asymptotes: y=±(a/b)x
Hyperbola x²/a² − y²/b² = 1 −aa F₁F₂ For a horizontal hyperbola, c²=a²+b² and the foci lie beyond the vertices.
Figure 8: Standard horizontal hyperbola with its asymptotes, vertices and foci.
Worked Example

For x²/16−y²/9=1, find the foci, eccentricity and asymptotes.

a=4, b=3, c²=16+9=25 ⇒ c=5.

Foci: (±5,0).

e=5/4.

y=±(3/4)x.

10. Quick Comparison of Standard Conics

ConicStandard equationKey parameter relationFocus / special feature
Circlex²+y²=a²radius = acentre (0,0)
Parabolay²=4axe=1focus (a,0), directrix x=−a
Ellipsex²/a²+y²/b²=1c²=a²−b²e=c/a<1
Hyperbolax²/a²−y²/b²=1c²=a²+b²e=c/a>1
Ellipse vs Hyperbola

This is a common exam error: for an ellipse, c²=a²−b²; for a hyperbola, c²=a²+b².

11. Worked Examples

Example 1: Identify a Circle

Find centre and radius of x²+y²−6x+8y−11=0.

Here 2g=−6⇒g=−3, 2f=8⇒f=4, c=−11.

Centre (3,−4).

r=√(9+16+11)=6.

Example 2: Check Tangency

Check whether 3x+4y=25 is tangent to x²+y²=25.

Distance from origin to line:

25/√(3²+4²)=25/5=5.

Radius is 5. Since distance = radius, the line is tangent.

Example 3: Tangent to a Circle

Find tangent to x²+y²=13 at (2,3).

2x+3y=13.

Example 4: Parabola Focus and Directrix

For y²=12x, identify the focus and directrix.

4a=12⇒a=3.

Focus: (3,0); directrix: x=−3.

Example 5: Tangent to a Parabola

Find tangent to y²=16x at (4,8).

a=4.

8y=8(x+4) ⇒ y=x+4.

Example 6: Normal to a Parabola

Find normal to y²=16x at (4,8).

Normal slope: −y₁/(2a)=−8/8=−1.

y−8=−(x−4) ⇒ x+y−12=0.

Example 7: Ellipse Standard Data

For x²/36+y²/20=1, find the foci.

a²=36, b²=20 ⇒ c²=16 ⇒ c=4.

Foci: (±4,0).

Example 8: Hyperbola Standard Data

For y²/9−x²/16=1, find the foci.

a²=9, b²=16 ⇒ c²=25 ⇒ c=5.

Since the positive term is y², the transverse axis is vertical. Foci: (0,±5).

Example 9: Tangent of Given Slope to a Circle

Find tangents of slope 3 to x²+y²=4.

Write y=3x+c. For tangency:

c²=4(1+9)=40.

c=±2√10.

Tangents: y=3x±2√10.

12. Problem-Solving Strategy

  1. Identify the conic from the equation and put it into standard form.
  2. For a circle, find the centre and radius before applying a distance-to-line tangency condition.
  3. For a tangent at a given point, first confirm that the point lies on the conic.
  4. For x²+y²=a², use xx₁+yy₁=a².
  5. For y²=4ax, use yy₁=2a(x+x₁).
  6. For a parabola normal, obtain the normal slope as the negative reciprocal of the tangent slope.
  7. For ellipse, use c²=a²−b²; for hyperbola use c²=a²+b².
  8. Look at which squared term has the larger or positive denominator to determine the principal axis.
  9. Sketch a quick diagram to verify orientation, signs and location of foci.

13. Common Mistakes and Warnings

Mistake 1: Wrong Circle Centre

For x²+y²+2gx+2fy+c=0, centre is (−g,−f).

Mistake 2: Tangency Without Distance

A tangent line must have perpendicular distance from the centre exactly equal to the radius.

Mistake 3: Point Not on Conic

Check the given point before using a tangent-at-point formula.

Mistake 4: Tangent Formula Mix-up

Circle: xx₁+yy₁=a²; parabola y²=4ax: yy₁=2a(x+x₁).

Mistake 5: Wrong Parabola Direction

The sign and squared variable determine whether the parabola opens right, left, up or down.

Mistake 6: Ellipse c² Formula

For an ellipse, c²=a²−b², not a sum.

Mistake 7: Hyperbola c² Formula

For a hyperbola, c²=a²+b².

Mistake 8: Confusing Normal with Radius for Parabola

Only for a circle does the normal automatically pass through the centre. A parabola has no centre.

14. Exam-Important Formula Sheet

Core Formulae
Circle: x²+y²=a² General circle: x²+y²+2gx+2fy+c=0 Circle tangent: xx₁+yy₁=a² Circle normal: x₁y−y₁x=0 Tangency of y=mx+c to x²+y²=a²: c²=a²(1+m²) Parabola: y²=4ax Parabola tangent at (x₁,y₁): yy₁=2a(x+x₁) Parametric tangent: ty=x+at² Parametric normal: y=−tx+2at+at³ Ellipse: x²/a²+y²/b²=1, c²=a²−b² Hyperbola: x²/a²−y²/b²=1, c²=a²+b²

15. Important Exam Questions

Short-Answer Questions

  1. Define a conic section.
  2. Write the centre and radius of a general circle.
  3. State the condition for y=mx+c to touch x²+y²=a².
  4. Write the tangent to x²+y²=a² at (x₁,y₁).
  5. Write the standard equation, focus and directrix of y²=4ax.
  6. Write the tangent and normal to y²=4ax at a given point.
  7. Write the standard equation of an ellipse and relation between a,b,c.
  8. Write the standard equation of a hyperbola and relation between a,b,c.

Derivation / Proof Questions

  1. Derive the condition c²=a²(1+m²) for tangency of y=mx+c to x²+y²=a².
  2. Derive the standard equation y²=4ax from the focus-directrix definition.
  3. Derive the tangent equation to y²=4ax at a given point.
  4. Obtain the equation of the normal to y²=4ax at a parametric point.
  5. Explain the standard forms of ellipse and hyperbola and identify their foci.

Numerical / Application Questions

  1. Find the centre and radius of a circle from its general equation.
  2. Determine whether a given line is tangent to a given circle.
  3. Find the value of a parameter for which a line becomes tangent to a circle.
  4. Find tangent and normal to a circle at a specified point.
  5. Find focus, directrix and latus rectum data for a standard parabola.
  6. Find tangent and normal to a parabola at a specified point.
  7. Determine the foci and eccentricity of a standard ellipse.
  8. Determine the foci, eccentricity and asymptotes of a standard hyperbola.

Diagram Questions

  1. Sketch a circle showing the radius perpendicular to a tangent.
  2. Sketch y²=4ax with focus and directrix.
  3. Sketch tangent and normal to a parabola at a point.
  4. Sketch a standard ellipse with foci and vertices.
  5. Sketch a standard hyperbola with foci, vertices and asymptotes.

16. One-Minute Revision

Quick Revision
  • A conic may be a circle, parabola, ellipse or hyperbola.
  • For x²+y²+2gx+2fy+c=0, centre is (−g,−f).
  • Radius is √(g²+f²−c).
  • A line is tangent to a circle when distance from centre to line equals radius.
  • For x²+y²=a², tangent at (x₁,y₁) is xx₁+yy₁=a².
  • The normal to a circle passes through the centre.
  • Standard right-opening parabola: y²=4ax.
  • Its focus is (a,0) and directrix is x=−a.
  • Parabola tangent: yy₁=2a(x+x₁).
  • At parametric point (at²,2at), tangent is ty=x+at².
  • Parametric normal is y=−tx+2at+at³.
  • Ellipse: x²/a²+y²/b²=1 with c²=a²−b².
  • Hyperbola: x²/a²−y²/b²=1 with c²=a²+b².
  • Ellipse has e<1; parabola e=1; hyperbola e>1.
  • Always use a quick sketch to check conic orientation and focus positions.

17. Diagram Practice

  • Four-conic comparison sketch.
  • Circle with centre, point of contact, radius and tangent.
  • Distance-from-centre tangency diagram.
  • Parabola y²=4ax with vertex, focus, directrix and latus rectum.
  • Tangent to a parabola at a point.
  • Tangent and normal to a parabola at the same point.
  • Standard ellipse with major axis, vertices and foci.
  • Standard hyperbola with vertices, foci and asymptotes.

Also Visit

Class 12 Mathematics Notes Class 12 Notes Practical File Question Papers Important Questions
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