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Terminology of Parabolas

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We now focus on some important terminology pertaining to parabolas. We will be using the parabola $${y^2} = 4ax$$ as illustration for this purpose; however, this discussion is general.

LATUS-RECTUM: This is a unique chord for a given parabola. It is the chord passing through the focus and perpendicular to the axis of the parabola.

The length of the latus rectum can easily be evaluated. Substituting x = a in the equation for the parabola gives two corresponding values for y, i.e., $$y = \pm 2a$$ corresponding to A and B. Thus, the length AB is 4a.

\Rightarrow \quad \boxed {{\begin{align}\begin{array}{l}{\rm{Length\;of \;Latus - Rectum \;of\;}}\\{y^2} = 4ax\end{array}{\;\;\;\;\;\;\; = \;\;\;\;4a}\end{align}}}

From now on, we will abbreviate the latus-rectum as LR. The length of the LR can also be seen to be 4a this way. Since A lies on the parabola, its distance from F, i.e., AF must equal its distance from x +a = 0, which is 2a by inspection. Thus, AB = 2AF = 4a.

FOCAL-CHORD: Any chord passing through the focus of the parabola will be its focal chord. The LR is a particular focal chord.

FOCAL DISTANCE: This, as the name suggests, is the distance of any point on the parabola from its focus, and which will by definition, by equal to its distance from the directrix.

Thus,

\Rightarrow\quad \boxed{{\begin{align}\begin{array}{l}{\rm{Focal\;distance\;of\;}}P(x,y)\\ {\rm{on\; }}{y^2} = 4ax\end{array}{\quad =\quad \left| {x + a} \right|}\end{align}}}

We now finally turn our attention to the most general form for the equation of a parabola, i.e., suppose that we are given an arbitrary fixed point F(h, k) and a fixed line $$L \equiv ax + by + c = 0$$ as the focus and directrix of the parabola. What is its equation is such a case?

We use the definition for the parabola, i.e. any point P(x, y) lying on it must be equidistant from F and L.

Referring to the figure above, we have

\begin{align}&\qquad \;\; P{M^2} = P{F^2}\\&\Rightarrow \quad\frac{{{{(ax + by + c)}^2}}}{{{a^2} + {b^2}}} = {(x - h)^2} + {(y - k)^2}\qquad\qquad\dots\left( 1 \right)\\& \Rightarrow \quad {a^2}{x^2} + {b^2}{y^2} + {c^2} + 2abxy + 2acx + 2bcy\\&\qquad \qquad \qquad \qquad \quad = ({a^2} + {b^2})\{ {x^2} + {y^2} + {h^2} + {k^2} - 2hx - 2ky\} \\&\Rightarrow \quad {b^2}{x^2} + {a^2}{y^2} - 2abxy - (2h({a^2} + {b^2}) + 2ac)x - (2k({a^2} + {b^2}) + 2bc)y\\&\qquad \qquad \qquad \qquad \quad + ({a^2} + {b^2})({h^2} + {k^2}) - {c^2} = 0 \end{align}

Whatever the coefficients maybe, we see that the equation of a parabola in general has the form

$A{x^2} + 2Hxy + B{y^2} + 2Gx + 2Fy + C = 0 \qquad \qquad \dots\left( 2 \right)$

So that (2) can be expressed in the form (1) (only then can (2) represent a parabola), it can be shown that the coefficients in (2) must satisfy the relations

\Delta = \left| {\begin{align}& A&&H&&G\\ &H&&B&&F\\& G&&F&&C \end{align}} \right| \ne 0 \;\;\;and \;\;{H^2} = AB

This is left to the rigor-bent reader as an exercise

Example – 2

Find the equation of the parabola with focus F(–1, –2) and directrix $$L \equiv x - 2y + 3 = 0.$$

Solution: Assuming P(x, y) to be any point on the parabola, we must have, by virtue of P being equidistant from F and L,

\begin{align}& \qquad \frac{{{{(x - 2y + 3)}^2}}}{5} = {(x + 1)^2} + {(y + 2)^2}\\\\& \Rightarrow \quad {x^2} + 4{y^2} + 9 - 4xy + 6x - 12y = 5{x^2} + 5{y^2} + 10x + 20y + 25\\\\ &\Rightarrow \quad 4{x^2} + {y^2} + 4xy + 4x + 32y + 16 = 0\end{align}

This is the required equation.

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