Meet the Ellipse Discover the shape behind orbits, tracks, and tunes.

Formal Definition

Ellipse

An ellipse is the set of all points in a plane whose distances to two fixed points, called foci, always add to the same constant value.

Which everyday objects do you think satisfy this constant-sum rule?

Focus on the Foci

https://asset.sparkl.ac/pb/sparkl-vector-images/img_ncert/8oN0MnbUnXZVXr4TFdkcc6w3isKruNZnGBX84Q9E.png

Ellipse with axes, centre and foci.

Key Parts on the Diagram

Label each part of the ellipse, then verify the focal rule.

Key Points:

  • A, B: ends of the major axis, length \(2a\).
  • C, D: ends of the minor axis, length \(2b\).
  • \(F_1, F_2\): foci, symmetric about centre O.
  • Any point \(P\) keeps \(PF_1 + PF_2 = 2a\).

The a² = b² + c² Link

\[a^{2}=b^{2}+c^{2}\]

Variable Definitions

a Semi-major axis length
b Semi-minor axis length
c Distance from centre to a focus

Applications

Locate Foci

Use \(c=\sqrt{a^{2}-b^{2}}\) to plot focus points quickly.

Find Eccentricity

Compute \(e=\frac{c}{a}\) to measure how “stretched” the ellipse is.

Verify Ellipse Data

Check if given \(a,b,c\) satisfy the relation before graphing.

Source: NCERT Class 11 Mathematics

Two Orientations

Major Axis Along x-axis

\( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1,\; a>b \)
Larger denominator under \(x\) → x-major orientation check
Foci at \((\pm c,0)\) with \(c^{2}=a^{2}-b^{2}\)
Vertices \((\pm a,0)\); major axis length \(2a\)

Major Axis Along y-axis

\( \frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}} = 1,\; a>b \)
Larger denominator under \(y\) → y-major orientation check
Foci at \((0,\pm c)\) with \(c^{2}=a^{2}-b^{2}\)
Vertices \((0,\pm a)\); major axis length \(2a\)

Key Similarities

Centre at \((0,0)\) for both orientations
Relation \(c^{2}=a^{2}-b^{2}\) and eccentricity \(e=\frac{c}{a}\)
Minor axis length \(2b\) and the condition \(a>b\)

Multiple Choice Question

Question

For the ellipse \( \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1 \), the point \( P(2,1) \) lies _____ the curve.

1
Inside the ellipse
2
On the ellipse
3
Outside the ellipse
4
Cannot be determined

Hint:

Substitute \( x = 2, y = 1 \). Compare the sum with 1 to decide the location.

Key Takeaways

Ellipses in a nutshell

Definition

Locus of points whose distances to two fixed foci add to a constant.

Standard Formulas

Centre at origin: \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1 \); focus distance \(c\) obeys \(c^{2}=a^{2}-b^{2}\).

Orientation

If \(a>b\), major axis lies on x-axis; if \(b>a\), on y-axis; rotation introduces an \(xy\) term.

Parameter Effects

Increasing \(a\) widens, \(b\) tallens; larger \(c\) raises eccentricity \(e=c/a\).