Question

Which of the following equations represents a parabola
A. $(x-y)^3=3$
B. $\dfrac{x}{y}-\dfrac{y}{x}=0$
C. $\dfrac{x}{y}-\dfrac{4}{x}=0$
D. None of these

Answer 1

Hint: A parabola is an equation of a curve in which a point on the curve is equally spaced from a fixed point and a fixed-line. The parabola's fixed point is referred to as the focus, and its fixed line is referred to as the directrix. To determine the correct equation form will compare the given equations with the standard equation of the parabola.

Formula Used: Standard equation of the parabola is:
$y^2=4ax$
$y^2=-4ax$
$x^2=4ax$
$x^2=-4ax$

Complete step by step solution: Let’s compare option (A) with the standard form of the parabola.
We have,
$(x-y)^3=3$
Here the raised power is 3. It does not represent an equation of the parabola. So option (A) is incorrect.

Now, compare option (b) with the standard form of the parabola.
On Simplifying we get,
$\dfrac{x}{y}-\dfrac{y}{x}=0$
$x^2=y^2$
Since it does not represent an equation of the parabola. So option (B) is incorrect.

Now, compare option (c) with the standard form of the parabola.
On Simplifying we get,
$\dfrac{x}{y}-\dfrac{4}{x}=0$
$x^2+4y=0$
$x^2=-4y$
Yes. It’s a parabola.

So, option C is correct.

Note: The parabola is an equation of a curve in which a point on the curve is equally spaced from a fixed point and a fixed line so carefully look at the standard and the general equation of the parabola.