Ellipse Geometry

01 / UNDERSTAND & APPLY

1. The focal sum fixes a bounded locus

An ellipse is the locus of points whose distances from two fixed foci have a constant sum. Place the foci at F₁=(−c,0), F₂=(c,0), with c≥0, and let the sum be 2a. A nondegenerate ellipse requires a>c. Equality would collapse the locus to the segment between the foci, while a<c is impossible by the triangle inequality.

Write b²=a²−c²>0. The standard horizontal-major-axis ellipse is x²/a²+y²/b²=1 with a≥b>0. The semi-major axis is a, the semi-minor axis is b and the focal distance from the center is c=√(a²−b²). Full axis lengths are 2a and 2b, not a and b.

When a=b, c=0 and the two foci coincide, giving a circle. This is a legitimate ellipse as a fixed-sum locus, but formulas involving division by eccentricity require a separate circle case. For a noncircular ellipse, 0<b<a and 0<c<a.

The equation immediately gives |x|≤a and |y|≤b. The vertices are (±a,0) and the co-vertices are (0,±b). These intercepts and symmetry axes provide quick checks on every later tangent or focal calculation.

02 / UNDERSTAND & APPLY

2. Derive the equation without forgetting distance signs

Let d₁=|PF₁| and d₂=|PF₂|. The definition is d₁+d₂=2a. Also d₁²−d₂²=(x+c)²+y²−[(x−c)²+y²]=4cx, so d₁−d₂=2cx/a.

Adding and subtracting the two relations gives d₁=a+(c/a)x and d₂=a−(c/a)x. Substitute the latter into d₂²=(x−c)²+y². Expanding and simplifying yields x²/a²+y²/(a²−c²)=1.

The reverse implication also matters because squaring can introduce extra points. On the standard ellipse, |x|≤a and c<a, so a±cx/a≥a−c>0. The squared distance identities therefore recover the positive distances d₁,d₂, whose sum is 2a. No extraneous branch remains.

For a vertical major axis, interchange x and y: x²/b²+y²/a²=1, with foci (0,±c). A translation replaces x,y by x−h,y−k. The larger denominator identifies the major-axis direction after the equation has been normalized to right side 1.

03 / UNDERSTAND & APPLY

3. The same ellipse has a focus–directrix description

The eccentricity is e=c/a, so 0≤e<1. For a noncircular horizontal ellipse, the directrices are x=±a/e=±a²/c. They lie outside the vertices because 1/e>1. They are not the lines through the foci.

For the right focus F₂=(c,0), the corresponding directrix is x=a/e. Every ellipse point has x≤a<a/e, so its distance to that line is a/e−x. Multiplying by e gives a−ex=d₂, the focal-distance expression derived earlier. This proves PF₂=e·distance(P,directrix).

The latus rectum through a focus is the chord perpendicular to the major axis. Setting x=c in the ellipse equation gives y=±b²/a, so its length is 2b²/a. The two endpoints share a focus as the midpoint only in the vertical coordinate; the focus is indeed the chord midpoint at (c,0).

As e increases toward 1 with a fixed, b=a√(1−e²) decreases and the ellipse becomes more elongated. As e→0 it approaches a circle; the directrices recede without bound rather than becoming finite lines through the center.

04 / UNDERSTAND & APPLY

4. The eccentric angle is a parameter, not usually the polar angle

The parametrization x=a cos θ,y=b sin θ satisfies the ellipse equation for every real θ and traces it once as θ runs through a full 2π interval. It is convenient because the normalized coordinates obey cos²θ+sin²θ=1.

The parameter θ is often called the eccentric angle. It is generally not the geometric angle the radius vector OP makes with the positive x-axis. Where both ratios are meaningful, tan of that geometric angle equals (b/a)tan θ, so they coincide only in special cases or for a circle.

For a translated ellipse, use x=h+a cos θ and y=k+b sin θ. Parameter differences of π produce opposite endpoints of a diameter through the center. Equal parameter increments do not in general correspond to equal arc lengths.

On the horizontal ellipse, the distances to the left and right foci are a+ex and a−ex. In parameter form these are a+c cos θ and a−c cos θ. Their sum remains 2a and both are positive on a nondegenerate ellipse.

05 / UNDERSTAND & APPLY

5. A tangent is a supporting line of a convex ellipse

At an ellipse point P=(x₁,y₁), the tangent equation is xx₁/a²+yy₁/b²=1. One derivation uses the gradient normal (x₁/a²,y₁/b²), which is nonzero on the ellipse. Another uses the unit-circle Cauchy–Schwarz inequality after scaling coordinates.

Indeed, for any ellipse point X, the normalized vectors (x/a,y/b) and (x₁/a,y₁/b) both have length 1. Their dot product is at most 1, with equality only when they coincide. Thus the line has exactly one contact point and all ellipse points lie on one side.

For P=(a cos θ,b sin θ), the tangent becomes x cos θ/a+y sin θ/b=1. At θ=0 or π it is vertical, x=±a; at θ=π/2 or 3π/2 it is horizontal, y=±b. These endpoint cases are included without dividing by x₁ or y₁.

A general line ux+vy=w with (u,v)≠(0,0) is tangent exactly when w²=a²u²+b²v². This follows because the linear expression ux+vy has extrema ±√(a²u²+b²v²) on the ellipse. A strict inequality w²<a²u²+b²v² gives two intersections; a larger square gives none.

06 / UNDERSTAND & APPLY

6. The normal is not usually the center-to-point line

The normal line at P is the line through P in the gradient direction (x₁/a²,y₁/b²). This direction is perpendicular to the tangent. A parametrization X=P+t(x₁/a²,y₁/b²) works at every ellipse point without exceptional slopes.

When x₁≠0, the normal slope is a²y₁/(b²x₁). The radius-vector slope is y₁/x₁. These are generally different unless a=b or the point lies on an axis. A familiar circle fact, that every normal passes through the center, therefore does not extend to a noncircular ellipse.

At a major vertex the normal is the major axis, and at a co-vertex it is the minor axis. Parameter-based normal formulas containing sec θ or csc θ may fail at precisely these easy points; the vector form handles them directly.

To find a normal through a given point, impose that the displacement to a candidate contact is parallel to the gradient and also impose the ellipse equation. This can produce several real contacts, so unlike tangents from an exterior point there is no general two-normal rule.

07 / UNDERSTAND & APPLY

7. Chord geometry is easiest in scaled coordinates

A horizontal chord y=k exists when |k|≤b and has endpoints x=±a√(1−k²/b²). Its length is 2a√(1−k²/b²). Equality |k|=b collapses it to a tangent contact; a genuine two-endpoint chord requires |k|<b.

For a chord with midpoint M=(x₀,y₀), write points on its line as M+td. The ellipse equation becomes a quadratic in t. Because the endpoint parameters are opposite about a midpoint, the linear coefficient must vanish: x₀dₓ/a²+y₀dᵧ/b²=0.

For a noncentral interior midpoint M, the chord line is xx₀/a²+yy₀/b²=x₀²/a²+y₀²/b². The midpoint must lie strictly inside for a nondegenerate chord. At the center both sides vanish and this equation becomes 0=0, reflecting that every diameter has the same midpoint and no unique chord direction is determined.

The midpoint chord and the tangent at a point use similar left sides but different right sides. The tangent uses 1 and assumes the point lies on the ellipse. The midpoint chord uses the normalized squared length of an interior point. Mixing these conditions changes the geometric object.

08 / UNDERSTAND & APPLY

8. Stretching a circle preserves some facts but not angles

The mapping (u,v)↦(au,bv) takes the unit circle to the ellipse. It multiplies area by ab, so the enclosed area is πab. It sends lines to lines and midpoints to midpoints, which helps explain chord and tangent formulas.

Unless a=b, this affine scaling does not preserve general angles or Euclidean lengths. A circle radius maps to a center-to-point segment, but that segment need not remain perpendicular to the mapped tangent. This is the geometric reason ellipse normals differ from circle radii.

For L=ux+vy on the ellipse, put x=a cos θ,y=b sin θ. Then L=au cos θ+bv sin θ, whose range is [−R,R] with R=√(a²u²+b²v²). Equality follows by choosing the unit vector (cos θ,sin θ) parallel or antiparallel to (au,bv).

This support calculation supplies exact distances between parallel tangents and extrema of linear expressions. It does not give extrema of arbitrary nonlinear functions without further work. Always identify which quantity is being optimized.

09 / UNDERSTAND & APPLY

9. Parallel chords and focal geometry reveal different bisectors

Consider chords parallel to a nonzero direction d=(u,v). The midpoint condition derived earlier is x₀u/a²+y₀v/b²=0. Thus their midpoints lie on a line through the center, a diameter of the ellipse. Conversely, an interior point on that diameter determines a chord parallel to d through the midpoint equation.

If the chords have finite nonzero slope m, take d=(1,m). Their midpoint diameter has slope −b²/(a²m). The product of the two slopes is −b²/a², not generally −1. For horizontal chords the midpoint diameter is vertical; for vertical chords it is horizontal. These exceptional cases are easier to see directly than by dividing by m.

Two diameter directions related this way are called conjugate directions. They arise by applying the ellipse scaling to perpendicular diameters of the unit circle. Because nonuniform scaling does not preserve angles, conjugate diameters need not be perpendicular. This is another place where a familiar circle picture must be interpreted carefully.

The foci produce a different angle relation. At an ellipse point P, let u₁ and u₂ be the unit vectors from the two foci toward P. The gradient of the distance sum is u₁+u₂, which is normal to the ellipse because the sum is constant along it. The two unit vectors have equal length, so their sum bisects their angle whenever they are not opposite.

Therefore the tangent makes equal angles with the two focal segments, giving the usual equal-angle reflection property. At a major vertex the focal directions coincide and the tangent is perpendicular to the axis; the limiting geometric statement remains consistent. This argument concerns tangent angles, not a claim that the normal passes through either focus or the center.

These constructions answer different questions: a midpoint diameter organizes an entire family of parallel chords, while the focal angle bisector identifies the local normal at one contact. A problem mentioning one of them should not be solved by substituting the formula for the other.

10 / UNDERSTAND & APPLY

10. Original practice with full solutions

Use a as the semi-major axis only after identifying the larger denominator. Keep focus, vertex and directrix positions distinct.

11 / UNDERSTAND & APPLY

11. Recover the picture before using a formula

Normalize the equation, identify the center and the larger denominator, and compute c²=a²−b². This establishes the orientation before any focus or tangent formula is used. A full axis length must be halved before being inserted as a semi-axis.

The focal sum is 2a, the eccentricity is below 1, the directrices lie outside the ellipse and the latus rectum has length 2b²/a. These geometric checks can catch a swapped axis or an incorrectly signed focal relation.

For tangents and normals, verify the contact point lies on the ellipse. Use the gradient to avoid zero-coordinate exceptions. For midpoint chords, distinguish an interior midpoint from a boundary contact and handle the center separately.

Finally remember what scaling preserves. It transforms a circle into an ellipse and scales area, but it does not preserve general angles. Keeping that distinction visible prevents the incorrect assumption that every ellipse normal goes through its center.