What is an Ellipse?

Ellipse

The locus of plane points whose distances to two fixed points (foci) add up to the same constant.

This is the formal locus definition of an ellipse.

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Standard Equation View

Ellipse graph for a=3, b=2

Example: \( \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1 \)

Every point \((x, y)\) on this curve obeys the standard equation.

Semi-major axis: \(a = 3\) units along the x-direction.

Semi-minor axis: \(b = 2\) units along the y-direction, giving a tighter vertical span.

Key Points:

  • Standard form links algebra to shape: \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \).
  • Semi-major axis \(a = 3\) → wider spread along x-axis.
  • Semi-minor axis \(b = 2\) → narrower stretch along y-axis.
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Focus–Directrix Insight

One focus and one line can generate every ellipse—here’s how.

1

Pick Focus & Directrix

Fix a point \(F\) and a non-intersecting line \(D\) in the plane.

2

Set Eccentricity \(e\)

Choose \(0 < e < 1\) and demand the ratio \(\\frac{PF}{PD}=e\) for all positions of \(P\).

3

Trace the Locus

The set of all points \(P\) satisfying the ratio forms a smooth closed curve—the ellipse.

Pro Tip:

Smaller \(e\) makes the ellipse more circular; as \(e\) approaches 1, it stretches out.

Equation Derivation

1
\[PF + PF' = 2a\]

By definition, an ellipse keeps the sum of distances from any point \(P\) to the foci constant \(2a\).

2
\[\sqrt{(x+c)^2 + y^2} + \sqrt{(x-c)^2 + y^2} = 2a\]

Insert coordinates \(F(-c,0)\), \(F'(c,0)\) and apply the distance formula to point \(P(x,y)\).

3
\[\sqrt{(x+c)^2 + y^2} = 2a - \sqrt{(x-c)^2 + y^2}\]

Isolate one radical to prepare for systematic squaring.

4
\[(x+c)^2 + y^2 = 4a^2 - 4a\sqrt{(x-c)^2 + y^2} + (x-c)^2 + y^2\]

Square both sides once, expand, and collect like terms.

5
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,\; b^2 = a^2 - c^2\]

After eliminating radicals and simplifying, divide by \(a^2b^2\) to obtain the standard form.

Key Insight:

Geometry meets algebra through \(b^2 = a^2 - c^2\); this link turns the distance-sum rule into the elegant equation \(x^2/a^2 + y^2/b^2 = 1\).

Multiple Choice Question

Question

Which statement is true for every point on an ellipse with foci \(F_{1}\) and \(F_{2}\)?

1
Distance to \(F_{1}\) equals distance to \(F_{2}\).
2
Sum of distances to \(F_{1}\) and \(F_{2}\) is constant.
3
Difference of distances to \(F_{1}\) and \(F_{2}\) is constant.
4
Product of distances to \(F_{1}\) and \(F_{2}\) is constant.

Hint:

Recall the geometric definition of an ellipse.

Ellipse vs Circle

Ellipse

Sum of distances to two foci is constant.
Axes lengths unequal, \(a \neq b\).
Eccentricity \(0 < e < 1\)  →  shows stretch.
Equation \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\).

Circle

All points at equal distance from one centre.
Axes lengths equal, \(a = b\).
Eccentricity \(e = 0\)  →  no stretch.
Equation \(x^{2} + y^{2} = r^{2}\).

Key Similarities

Closed conic sections from slicing a right circular cone.
Symmetric about both the major and minor axes.
Represented by second-degree equations in \(x\) and \(y\).

Key Takeaways

Foci Anchor Shape

An ellipse is all points whose distances to two foci add to a constant.

Equation Reveals Geometry

Standard form \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1 \) turns geometric distances into algebraic balance.

Parameters Shape Size

Semi-axes \(a, b\) set width and height; eccentricity \(e=\sqrt{1-\frac{b^{2}}{a^{2}}}\) gauges flatness.