Find the standard form of the equation of the parabola with the given characteristic and vertex at the origin. Focus:

Question

Quizplus · Accepted Answer

Since the vertex is at the origin, the equation of the parabola is of the form $y=ax^2$. The focus is given as $(0,\frac{1}{4a})$. Using the definition of the focus and the distance formula, we get:

$$\sqrt{(0-x)^2+(\frac{1}{4a}-y)^2}=| \frac{1}{4a} - y|$$

Squaring both sides, we get:

$$x^2+(y-\frac{1}{4a})^2=(\frac{1}{4a}-y)^2$$

Expanding and simplifying, we get:

$$4ay=1$$

Therefore, the equation of the parabola is $y=\frac{1}{4a}x^2$ or $4ay=x^2$. The answer is (D).

Find the Standard Form of the Equation of the Parabola