IIT JEE Main Maths -Unit 10- Standard equation and parameters (center, vertices, foci, eccentricity)- Study Notes-New Syllabus
IIT JEE Main Maths -Unit 10- Standard equation and parameters (center, vertices, foci, eccentricity)- Study Notes – New syllabus
IIT JEE Main Maths -Unit 10- Standard equation and parameters (center, vertices, foci, eccentricity)- Study Notes -IIT JEE Main Maths – per latest Syllabus.
Key Concepts:
- Ellipse: Standard Equation and Parameters
- Ellipse: Latus Rectum
- Ellipse: Parametric Form
Ellipse: Standard Equation and Parameters
An ellipse is the set of all points whose sum of distances from two fixed points (foci) is constant.
Standard ellipse forms depend on whether the major axis is horizontal or vertical.
1. Standard Equation of Ellipse
(A) Horizontal Major Axis
\( \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1,\quad a > b > 0 \)
(B) Vertical Major Axis
\( \dfrac{x^2}{b^2} + \dfrac{y^2}{a^2} = 1,\quad a > b > 0 \)
In both forms:
\( a = \) semi-major axis
\( b = \) semi-minor axis
\( c = \) distance of focus from center
Key relation:
\( c^2 = a^2 – b^2 \)
2. Parameters of Ellipse
| Parameter | Horizontal Ellipse | Vertical Ellipse |
| Center | \( (0,0) \) | \( (0,0) \) |
| Vertices | \( (\pm a,0) \) | \( (0,\pm a) \) |
| Co-vertices | \( (0,\pm b) \) | \( (\pm b,0) \) |
| Foci | \( (\pm c,0) \) | \( (0,\pm c) \) |
| Eccentricity | \( e = \dfrac{c}{a} \) | \( e = \dfrac{c}{a} \) |
3. Eccentricity of Ellipse
Eccentricity (measure of flatness):
\( e = \dfrac{c}{a},\quad 0 < e < 1 \)
Relation with a and b:
\( b^2 = a^2(1 – e^2) \)
4. Lengths
- Major axis length: \( 2a \)
- Minor axis length: \( 2b \)
- Distance between foci: \( 2c \)
5. Auxiliary Circle
The auxiliary circle of the ellipse \( \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \) is:
\( x^2 + y^2 = a^2 \)
Useful for parametric form.
6. Parametric Coordinates
For ellipse \( \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \):
\( (x,y) = (a\cos\theta,\ b\sin\theta) \)
Used heavily in tangent and normal problems.
Example
For ellipse \( \dfrac{x^2}{25} + \dfrac{y^2}{9} = 1 \), find its eccentricity.
▶️ Answer / Explanation
\( a^2 = 25,\ b^2 = 9 \)
\( c^2 = a^2 – b^2 = 25 – 9 = 16 \)
\( c = 4 \)
\( e = \dfrac{c}{a} = \dfrac{4}{5} \)
Answer: \( e = \dfrac{4}{5} \)
Example
Find center, vertices, and foci of ellipse \( \dfrac{x^2}{16} + \dfrac{y^2}{4} = 1. \)
▶️ Answer / Explanation
Here \( a^2 = 16,\ b^2 = 4 \) → major axis horizontal.
Center: \( (0,0) \)
Vertices: \( (\pm 4,0) \)
\( c^2 = 16 – 4 = 12 \Rightarrow c = 2\sqrt{3} \)
Foci: \( (\pm 2\sqrt{3}, 0) \)
Example
If the eccentricity of an ellipse is \( \dfrac{3}{5} \) and its minor axis length is 8, find the length of its major axis.
▶️ Answer / Explanation
Given:
\( e = \dfrac{3}{5},\ 2b = 8 \Rightarrow b = 4 \)
Use relation:
\( b^2 = a^2(1 – e^2) \)
\( 16 = a^2\left(1 – \dfrac{9}{25}\right) = a^2 \cdot \dfrac{16}{25} \)
Solve for a²:
\( a^2 = 25 \)
So a = 5.
Major axis length = 2a = 10
Ellipse: Latus Rectum
The latus rectum is a chord through a focus, perpendicular to the major axis.
For ellipse:
\( \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \)
1. Coordinates of Foci
Foci lie at:
\( (\pm c,0) \), where \( c^2 = a^2 – b^2 \)
2. Equation of Latus Rectum
Since latus rectum is perpendicular to major axis (horizontal), its equation through right focus is:
\( x = c \)
and for left focus:
\( x = -c \)
3. Endpoints of Latus Rectum
Put \( x = c \) into the ellipse:
\( \dfrac{c^2}{a^2} + \dfrac{y^2}{b^2} = 1 \)
Since \( c^2 = a^2 – b^2 \):
\( \dfrac{a^2 – b^2}{a^2} + \dfrac{y^2}{b^2} = 1 \)
\( \Rightarrow \dfrac{y^2}{b^2} = \dfrac{b^2}{a^2} \)
\( y = \pm \dfrac{b^2}{a} \)
Endpoints:
\( (c,\ \pm \dfrac{b^2}{a}) \)
Similarly for left focus:
\( (-c,\ \pm \dfrac{b^2}{a}) \)
4. Length of Latus Rectum
The distance between endpoints:
\( \text{LLR} = 2 \cdot \dfrac{b^2}{a} \)
This is true for both sides.
5. For Vertical Ellipse
For ellipse:
\( \dfrac{x^2}{b^2} + \dfrac{y^2}{a^2} = 1 \)
foci are \( (0,\pm c) \), and latus rectum is horizontal through each focus:
\( y = \pm c,\quad \text{endpoints} = \left(\pm\dfrac{b^2}{a},\ c\right) \)
Length is still:
\( \text{LLR} = 2\dfrac{b^2}{a} \)
Example
Find the length of latus rectum of the ellipse \( \dfrac{x^2}{25} + \dfrac{y^2}{9} = 1. \)
▶️ Answer / Explanation
\( \text{LLR} = 2\dfrac{b^2}{a} = 2\dfrac{9}{5} = \dfrac{18}{5} \)
Answer: \( \dfrac{18}{5} \)
Example
Find endpoints of latus rectum of ellipse \( \dfrac{x^2}{36} + \dfrac{y^2}{16} = 1. \)
▶️ Answer / Explanation
\( a^2 = 36 \Rightarrow a=6,\quad b^2 = 16 \Rightarrow b=4 \)
\( c^2 = a^2 – b^2 = 20 \Rightarrow c = 2\sqrt{5} \)
Endpoints:
\( (c,\ \pm\dfrac{b^2}{a}) = \left(2\sqrt{5},\ \pm\dfrac{16}{6}\right) = \left(2\sqrt{5},\ \pm\dfrac{8}{3}\right) \)
Example
The latus rectum of an ellipse has length 10 and its eccentricity is \( \dfrac{4}{5} \). Find the semi-major axis \( a \).
▶️ Answer / Explanation
LLR = \( 2\dfrac{b^2}{a} = 10 \)
\( b^2 = a^2(1-e^2) = a^2\left(1 – \dfrac{16}{25}\right) = a^2 \cdot \dfrac{9}{25} \)
Put in LLR formula:
\( 10 = 2\dfrac{(9/25)a^2}{a} = \dfrac{18}{25}a \)
\( a = \dfrac{250}{18} = \dfrac{125}{9} \)
Answer: \( a = \dfrac{125}{9} \)
Ellipse: Parametric Form
Parametric form is one of the most powerful tools for solving tangent, normal, and locus problems in ellipses. For ellipse:
\( \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \)
1. Standard Parametric Coordinates
Every point on the ellipse can be written as:
\( (x, y) = (a\cos\theta,\ b\sin\theta) \)
Here \( \theta \) is called the eccentric angle. This representation automatically satisfies the ellipse equation.
2. Why Parametric Form Works
Substitute \( x = a\cos\theta,\ y = b\sin\theta \):
\( \dfrac{a^2\cos^2\theta}{a^2} + \dfrac{b^2\sin^2\theta}{b^2} = \cos^2\theta + \sin^2\theta = 1 \)
Hence all such points lie on the ellipse.
3. Parameter Range
\( \theta \in [0, 2\pi] \) covers the entire ellipse exactly once.
4. Parametric Coordinates for Vertical Ellipse
For ellipse:
\( \dfrac{x^2}{b^2} + \dfrac{y^2}{a^2} = 1 \)
coordinates are:
\( (x, y) = (b\cos\theta,\ a\sin\theta) \)
5. Useful Parametric Results
- Distance from center: \( \sqrt{a^2\cos^2\theta + b^2\sin^2\theta} \)
- Distance from focus (for horizontal major axis):
\( a – e x = a – e(a\cos\theta) = a(1 – e\cos\theta) \) - Slope of tangent at parametric point: \( m = -\dfrac{b\cos\theta}{a\sin\theta} \)
Example
Find parametric coordinates of the point on the ellipse \( \dfrac{x^2}{25} + \dfrac{y^2}{9} = 1 \) when \( \theta = \dfrac{\pi}{6} \).
▶️ Answer / Explanation
\( a = 5,\ b = 3 \)
\( x = a\cos\theta = 5 \cdot \dfrac{\sqrt{3}}{2} = \dfrac{5\sqrt{3}}{2} \)
\( y = b\sin\theta = 3 \cdot \dfrac{1}{2} = \dfrac{3}{2} \)
Answer: \( \left(\dfrac{5\sqrt{3}}{2},\ \dfrac{3}{2}\right) \)
Example
Find the eccentric angle \( \theta \) of point \( (4,3) \) on ellipse \( \dfrac{x^2}{16} + \dfrac{y^2}{9} = 1. \)
▶️ Answer / Explanation
\( a=4,\ b=3 \)
\( 4 = 4\cos\theta \Rightarrow \cos\theta = 1 \Rightarrow \theta = 0 \)
Check y:
\( y = 3\sin\theta = 0 \neq 3 \)
→ Try y coordinate:
\( 3 = 3\sin\theta \Rightarrow \sin\theta = 1 \Rightarrow \theta = \dfrac{\pi}{2} \)
Check x:
\( x = 4\cos\theta = 0 \neq 4 \)
The point \( (4,3) \) does not lie on the ellipse.
No real eccentric angle exists.
Example
Find all eccentric angles of points on ellipse \( \dfrac{x^2}{25} + \dfrac{y^2}{9} = 1 \) for which \( x = y \).
▶️ Answer / Explanation
Parametric: \( x = 5\cos\theta,\ y = 3\sin\theta \)
Given: \( 5\cos\theta = 3\sin\theta \)
\( \tan\theta = \dfrac{5}{3} \)
Thus angles:
\( \theta = \tan^{-1}\left(\dfrac{5}{3}\right) \)
and \( \theta = \pi + \tan^{-1}\left(\dfrac{5}{3}\right) \)
These are the required eccentric angles.
Notes and Study Materials
- Concepts of Ellipse
- Ellipse Master File
- Ellipse Revision Notes
- Ellipse Formulae
- Ellipse Reference Book
- Ellipse Past Many Years Questions and Answer
Examples and Exercise
IIT JEE (Main) Mathematics ,”Ellipse” Notes ,Test Papers, Sample Papers, Past Years Papers , NCERT , S. L. Loney and Hall & Knight Solutions and Help from Ex- IITian
About this unit
Ellipse
IITian Academy Notes for IIT JEE (Main) Mathematics – Ellipse
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IIT JEE (Main) Mathematics, Ellipse Solved Examples and Practice Papers.
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Hall & Knight IIT JEE (Main) Mathematics
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