Question

Find the differential equations of all parabolas each having latus rectum $4a$ and whose axes are parallel to the x-axis.

• A. $x{\left(\frac{dy}{dx}\right)}^2=a$
• B. $x{\left(\frac{dy}{dx}\right)}^2=-a$
• C. $x{\left(\frac{dy}{dx}\right)}^2=2a$
• D. $x{\left(\frac{dy}{dx}\right)}^2=-2a$

Answer 1

The correct option is A $x{\left(\frac{dy}{dx}\right)}^2=a$

The equation of the parabola is ${(y-k)}^2=4ax\cdots \left(1\right)$
$\therefore 2(y-k)y_1=4a$
or $y-k=\frac{2a}{y_1}$
Putting in (1), we get
$\frac{4a^2}{y_1^2}=4ax$ or $x{\left(\frac{dy}{dx}\right)}^2=a$