The Parabola: Quick Revision

A brief note on The Parabola. Read it before a test. Try each question at the end before you open its answer.

The curve

• Definition: $PF = PM$ for a focus $F$ and a directrix $l$. The eccentricity is $e = 1$.
• Standard forms: $y^2 = 4ax$ (focus $(a, 0)$, directrix $x = -a$), $y^2 = -4ax$, $x^2 = 4ay$ and $x^2 = -4ay$.
• Latus rectum: length $4a$, ends $(a, \pm 2a)$ for $y^2 = 4ax$.
• Focal distance: $x_1 + a$ for a point $(x_1, y_1)$ on $y^2 = 4ax$.
• Parametric form: $(at^2, 2at)$.
• Shifted form: $(y - k)^2 = 4a(x - h)$ has the vertex $(h, k)$.

Tangents and normals

• Tangent at $(x_1, y_1)$: $yy_1 = 2a(x + x_1)$. At $t$: $ty = x + at^2$.
• Tangent with slope $m$: $y = mx + \dfrac{a}{m}$, touching at $\left(\dfrac{a}{m^2}, \dfrac{2a}{m}\right)$.
• Normal at $t$: $y + tx = 2at + at^3$. With slope $m$: $y = mx - 2am - am^3$.
• Subtangent and subnormal: $2at^2$ and $2a$.
• Perpendicular tangents meet on the directrix. Normals through a point: up to $3$, from $am^3 + (2a - h)m + k = 0$.

Focal chords and reflection

• Ends: $t$ and $-\dfrac{1}{t}$. Length: $a\left(t + \dfrac{1}{t}\right)^2$.
• Harmonic property: $\dfrac{1}{SP} + \dfrac{1}{SQ} = \dfrac{1}{a}$.
• Tangents at the ends are perpendicular and meet on the directrix.
• Reflection: a ray parallel to the axis goes through the focus, since $FT = FP$.

A worked example

Find the tangent to $y^2 = 8x$ at $(2, 4)$, and the normal there.

1. Check. $16 = 8 \times 2$. Here $a = 2$.
2. Tangent. $4y = 4(x + 2)$, so $y = x + 2$.
3. Normal. The slope is $-\dfrac{y_1}{2a} = -1$, so $y - 4 = -(x - 2)$, which is $x + y = 6$.

Practice with new numbers

A program to try

The program checks the tangent and the normal at a point and then finds the focal chord through the same point.

Mistakes to avoid

• $a$ read as the coefficient. For $y^2 = 12x$ the value is $a = 3$.
• Tangent slope. It is $\dfrac{2a}{y_1}$ or $\dfrac{1}{t}$, not $\dfrac{y_1}{2a}$.
• Focal chord. The product $t_1t_2$ is $-1$.
• Directrix. It is on the other side of the vertex from the focus.

Check yourself

1. Find the focus and the directrix of $y^2 = 20x$.

2. Find the tangent to $y^2 = 4x$ with slope $2$.

3. Find the length of the focal chord of $y^2 = 4x$ that has one end at $t = 3$.

4. Where do perpendicular tangents to a parabola meet?

5. What is the subnormal of $y^2 = 12x$?

Sources

• NCERT, Mathematics for Class XI, Chapter 10 (Conic Sections), the section on the parabola.
• OpenStax, Precalculus 2e, Chapter 12 (Analytic Geometry), the section on the parabola.