Answered: For each part, find the equation of the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

For each part, find the equation of the parabola, assuming the vertex is at the origin.
Then sketch the graph labeling the focus and directrix:
a. Contains (3, 4) and has y-axis symmetry
b. Has focus (0, −2)
c. Contains the points (2, 4) and (2, −4)?

Expert Solution

Step 1

Let us assume that in each case, vertex is at $(0, 0)$

Step 2

a. The given parabola contains the point $(3, 4)$ and axis of symmetry is $y -$ axis.

as vertex is at $(0, 0)$ and axis of symmetry is $y -$ axis, the equation of parabola will have form

$y = a x^{2}$ where $a$ is constant

As $(3, 4)$ is a point on the parabola, it will satisfy the equation of parabola

we have

$4 = a {(3)}^{2}$

$\Rightarrow 4 = 9 a$

$\Rightarrow a = \frac{4}{9}$

Also, focus of the parabola is $(0, \frac{1}{4 a})$

i.e. $(0, \frac{1}{4 \times \frac{4}{9}}) = (0, \frac{9}{16})$

Also, as axis of symmetry is $y$ axis and vertex is at origin,

then if focus is at $(0, b)$ , then equation of directrix is $y = - b$

Here focus is at point $(0, \frac{9}{16})$

Hence, equation of directrix is $y = - \frac{9}{16}$

We draw this parabola along with its directrix

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