Parabola

A conic section is defined locus-wise as the path tracked by a variable coordinate point $P$ whose distance from a fixed point $S$ (the focus) bears a constant ratio $e$ (the eccentricity) to its perpendicular distance from a fixed straight line $L = 0$ (the directrix). \[ \frac{\text{Distance from Focus } (SP)}{\text{Distance from Directrix } (PM)} = e \implies SP = e \cdot PM. \]

\[ \begin{array}{ll} \textbf{Eccentricity Value } (e) & \textbf{Resulting Conic Profile} \\ e = 1 & \text{Parabola} \\ 0 < e < 1 & \text{Ellipse} \\ e > 1 & \text{Hyperbola} \\ e = 0 & \text{Circle} \\ \end{array} \]

\begin{formulabox}[Standard Form and Key Elements ($a > 0$)] \[ y^2 = 4ax \]

When the vertex of the parabola is translated away from the origin to a new coordinate point $(h, k)$, the standard equations shift linearly: \[ (y - k)^2 = 4a(x - h) \quad \text{or} \quad (x-h)^2 = 4a(y-k). \]

[leftmargin=*]
• Focal Distance of Parameter $t$: The distance from the focus $S(a,0)$ to the parameter point $P(t)$ simplifies to: \[ |SP| = at^2 + a = a(1+t^2). \]
• Slope of a General Chord: The straight line connecting two distinct parameter points $P(t_1)$ and $Q(t_2)$ has its slope calculated as: \[ m = \frac{2at_2 - 2at_1}{at_2^2 - at_1^2} = \frac{2a(t_2 - t_1)}{a(t_2 - t_1)(t_2 + t_1)} = \frac{2}{t_1 + t_2} \quad (t_1 + t_2 \neq 0). \]

\begin{formulabox}[Point Form] \[ y y_1 = 2a(x + x_1) \] Derived using the standard $T = 0$ transform configuration matrix template: replace $y^2 \to y y_1$, and linear components $x \to \frac{x+x_1}{2}$. \end{formulabox}

\begin{formulabox}[Parametric Form — at point $t$] \[ ty = x + at^2 \] \end{formulabox}

\begin{formulabox}[Slope Form — slope $m$] \[ y = mx + \frac{a}{m} \quad (m \neq 0) \] The matching precise point of tangency contact is given by: $\left(\dfrac{a}{m^2},\, \dfrac{2a}{m}\right)$. \end{formulabox}

\begin{importantbox}[Condition of Tangency] The straight line $y = mx + c$ is a valid tangent line to the standard parabola $y^2 = 4ax$ if and only if: \[ c = \frac{a}{m}. \] \end{importantbox}

1. Tangents drawn at the endpoints of the latus rectum intersect at right angles on the directrix line.
2. The point of intersection of two tangents drawn at parameters $t_1$ and $t_2$ is given by: $\bigl(at_1t_2,\, a(t_1+t_2)\bigr)$.
3. If the tangents at $t_1$ and $t_2$ intersect at right angles ($90^\circ$), then $t_1 t_2 = -1$.

\begin{formulabox}[Point Form] \[ y - y_1 = -\frac{y_1}{2a}(x - x_1) \] \end{formulabox}

\begin{formulabox}[Parametric Form — at parameter $t$] \[ y + tx = 2at + at^3 \] \end{formulabox}

\begin{formulabox}[Slope Form — slope $m$] \[ y = mx - 2am - am^3 \] The matching base foot of the normal vector evaluates coordinate-wise to: $\bigl(am^2,\, -2am\bigr)$. \end{formulabox}

For a general external coordinate point $(h, k)$, the slope-form normal equation generates a cubic equation in terms of $m$: \[ am^3 + m(2a - h) + k = 0. \] Because it is a cubic equation, a maximum of three real normals can be drawn from an external point to a parabola.

\begin{importantbox}[Condition for Three Real Normals] Three distinct real normals can be drawn from an external point $(h, k)$ to the parabola $y^2 = 4ax$ if and only if: \[ h > 2a \quad \text{and} \quad k^2 < \frac{4}{27}\cdot\frac{(h-2a)^3}{a}. \] \end{importantbox}

If the three normals drawn at parameters $t_1, t_2,$ and $t_3$ are concurrent at a single point $(h,k)$, these parameters are called co-normal points. They satisfy the following properties derived from Vieta's formulas: \[ t_1 + t_2 + t_3 = 0, \quad t_1 t_2 + t_2 t_3 + t_3 t_1 = \frac{2a-h}{a}, \quad t_1 t_2 t_3 = -\frac{k}{a}. \]

A chord passing through the focus $(a, 0)$ is called a focal chord.

• The semi-latus rectum is the harmonic mean of the focal segments: $\dfrac{1}{SP} + \dfrac{1}{SQ} = \dfrac{1}{a}$.
• Total length of a focal chord $= a\left(t - \dfrac{1}{t}\right)^2 = a\left(t + \dfrac{1}{t}\right)^2$, where $t = t_1$.
• The minimum length of a focal chord is $4a$ (the latus rectum).

If a chord is bisected exactly at an interior point $M(h, k)$, its equation can be found using the $T = S_1$ formula: \begin{formulabox}[Equation of Chord with Midpoint $(h,k)$] \[ ky - 2a(x + h) = k^2 - 4ah \quad \Longleftrightarrow \quad T = S_1 \] where $T \equiv yy_1 - 2a(x+x_1)$ and $S_1 \equiv y_1^2 - 4ax_1$. \end{formulabox}

The straight line connecting the two points of tangency where tangents drawn from an external point $P(x_1, y_1)$ touch the curve is called the chord of contact. \begin{formulabox}[Chord of Contact: $T = 0$] \[ y y_1 = 2a(x + x_1) \] \end{formulabox}

% ============================================================

\begin{formulabox}[Polar of a Point $(x_1, y_1)$ w.r.t. $y^2 = 4ax$] \[ y y_1 = 2a(x + x_1) \quad \Longleftrightarrow \quad T = 0 \] The point $(x_1, y_1)$ is called the pole, and the resulting line equation is its polar. \end{formulabox}

\begin{property} Conjugate Points Property: If the polar line of a point $A$ passes through another point $B$, then the polar line of $B$ is guaranteed to pass through $A$. \end{property}

% ============================================================

For the parabola $y^2 = 4ax$, geometric projections of the tangent and normal lines onto the axis of symmetry from any point $P(x_1, y_1)$ yield constant length properties:

\begin{formulabox}[Subtangent and Subnormal Lengths] \[ \text{Length of Subtangent} = 2x_1, \qquad \text{Length of Subnormal} = 2a \] \end{formulabox}

The length of the subnormal is constant and equal to the semi-latus rectum $2a$ for all points on the parabola.

% ============================================================

\begin{importantbox}[Reflection Property] An incoming ray travelling parallel to the axis of symmetry of a parabola passes through the focus after reflecting off the parabolic surface curve.

1. The focal radius line $SP$, and
2. The horizontal line through $P$ drawn parallel to the axis of symmetry.

% ============================================================

\begin{examplebox}[Common Tangents: $y^2 = 4ax$ and $x^2 = 4by$] Let the required common tangent have a slope $m$. \[ y = mx + \frac{a}{m} \quad \text{--- (Tangent equation template for } y^2=4ax \text{)} \] Now rearrange the standard vertical parabola tangent layout to match slope form: \[ x = \frac{y}{m} - bm^2 \implies y = mx + bm^3 \quad \text{--- (Tangent equation template for } x^2=4by \text{)} \] Equate the two expressions for the y-intercept constant to solve for $m$: \[ \frac{a}{m} = bm^3 \implies m^4 = \frac{a}{b} \implies m = \left(\frac{a}{b}\right)^{1/4}. \] \end{examplebox}

% ============================================================

1. The tangent drawn at a parameter point $t$ intersects the tangent at the vertex ($x=0$) at the coordinate point $(0, at)$.
2. The orthocenter of the triangle formed by three tangent lines drawn to a parabola always lies on the directrix line curve.
3. The circumcircle of the triangle formed by three tangents drawn to a parabola always passes through the focus $S(a,0)$.
4. If the normal line drawn at parameter $t$ intersects the parabola again at a new parameter point $t'$, their relationship is given by: \[ t' = -t - \frac{2}{t}. \]
5. Director Circle Property: A parabola has no director circle curve. The locus of the point of intersection of perpendicular tangents is the directrix line ($x = -a$).
6. The area of a triangle inscribed inside the parabola $y^2 = 4ax$ with vertices at parameter points $t_1, t_2,$ and $t_3$ is: \[ \Delta = \frac{a^2}{2} |t_1 - t_2||t_2 - t_3||t_3 - t_1|. \]

% ============================================================

1. Find the angle subtended by the double ordinate of length $8a$ at the vertex of the standard parabola $y^2 = 4ax$.
2. If $P$ and $Q$ represent the parameter endpoints of a focal chord of the parabola $y^2 = 4ax$ and $S$ is the focus, prove that $\dfrac{1}{SP} + \dfrac{1}{SQ} = \dfrac{1}{a}$.
3. The normal line at a point $P$ on the parabola $y^2 = 4ax$ intersects the curve again at a point $Q$. Prove that the locus of the midpoint of the chord segment $PQ$ is given by the curve equation $y^2(y^2 + 2a^2) + 8a^4 = 0$. [JEE Advanced]
4. Tangent lines are drawn from the external point $(-1, 2)$ to the parabola $y^2 = 4x$. Find the equation of the chord of contact and calculate its length.
5. Three distinct real normals can be drawn to the parabola $y^2 = 4x$ through the point $(c, 0)$ lying on its axis of symmetry. Show that the parameter constant must satisfy $c > 2$.

1. The angle subtended by the double ordinate of length $8a$ at the vertex of $y^2 = 4ax$ evaluates to:
   1. $90^\circ$
   2. $\tan^{-1}(2)$
   3. $2\tan^{-1}(2)$
   4. $45^\circ$
2. If the chord of contact from $(-1, 2)$ to $y^2 = 4x$ has length $L$, the value of $L$ matches:
   1. $4$
   2. $2\sqrt{2}$
   3. $4\sqrt{2}$
   4. $8$
3. Find the range of $c$ for which three distinct real normals can be drawn to $y^2 = 4x$ from $(c, 0)$:
   1. $c > 2$
   2. $c < 2$
   3. $c > 4$
   4. $c > 1$
4. If a focal chord has segments $SP=4$ and $SQ$, and $a=2$, find the length of the segment $SQ$:
   1. $4$
   2. $2$
   3. $1$
   4. $8$
5. The locus of the point of intersection of perpendicular tangents to $y^2 = 16x$ is:
   1. $x + 4 = 0$
   2. $x - 4 = 0$
   3. $y = 4$
   4. $x = 0$

% ============================================================

% ============================================================

\begin{formulabox}[All Parabola Formulas at a Glance] Standard Parabola Layout Format: $y^2 = 4ax$; $\;a = $ distance from vertex to focus.

\medskip