"A latus rectum of a conic section is..." was answered by our community

preprekomW

Answered question

2021-03-05

A latus rectum of a conic section is a chord through a focus parallel to the directrix. Find the area bounded by the parabola

y = x^{2} / (4 c)

and its latus rectum.

FieniChoonin

Skilled2021-03-06Added 102 answers

Step 1
It is known that the area bounded by the curves

y = f (x) and y = g (x) o n [a, b]

is given by

A = \int_{a}^{b}

(upper curve - lower curve) dx.
From the figure, the equation of latus rectum is

y = c

which is upper curve.
Note that the given graph is about y-axis.
Step 2
Substitute

y = c \in y = \frac{x^{2}}{4 c}

and obtain that

x = 2 x

Thus, the area bounded can be computed as follows.

A = 2 \int_{0}^{2 c}

(upper curve - lower curve) dx

= 2 \int_{0}^{2 c} (c - \frac{x^{2}}{4 c}) d x

= 2 {[c x - \frac{x^{3}}{12 c}]}_{0}^{2 c}

= 2 [2 c^{2} - \frac{2}{3} c^{2}]

= \frac{8}{3} c^{2}

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