Suggested languages for you:

Americas

Europe

Problem 757

# Prove that if $$\mathrm{F}(0,1)$$ is the focus, and the line $$\mathrm{y}=-1$$ is the directrix, then the equation of the parabola is $\mathrm{y}=(1 / 4) \mathrm{x}^{2}$.

Expert verified
To prove that the given parabola has the focus F(0,1) and directrix y = -1, we define a general point P(x,y) on the parabola and find the distances PF and PD between the point and the focus, and the point and the directrix, respectively. Using the definition of a parabola, we equate the two distances, square both sides, and simplify the equation to obtain $$x^2 = 4y$$. Dividing by 4, we get the final equation of the parabola as $$y = \frac{1}{4}x^2$$.
See the step by step solution

## Step 1: Define a general point on the parabola

Let P(x,y) be a general point on the parabola.

## Step 2: Find the distance between the point and the focus

Using the distance formula: $$PF = \sqrt{(x-0)^2 + (y-1)^2} = \sqrt{x^2 + (y-1)^2}$$.

## Step 3: Find the distance between the point and the directrix

Since the directrix is a horizontal line at y=-1, the vertical distance between the point P(x,y) and the directrix is given by: $$PD = y - (-1) = y + 1$$.

## Step 4: Equate the two distances

According to the definition of a parabola, the distance between the point and the focus must be equal to the distance between the point and the directrix. Therefore, we have: $$\sqrt{x^2 + (y-1)^2} = y + 1$$.

## Step 5: Simplify the equation

To get rid of the square root, square both sides of the equation: $$(x^2 + (y-1)^2) = (y + 1)^2$$. Expand both sides: $$x^2 + y^2 - 2y + 1 = y^2 + 2y + 1$$. Now, notice that y^2 and 1 terms cancel out and simplify the equation to obtain the equation of the parabola: $$x^2 = 4y$$.

## Step 6: Divide by 4 to get the final equation

To get the final equation of the parabola, divide both sides by 4: $$y = \frac{1}{4}x^2$$. Thus, we have proved that if F(0,1) is the focus and the line y = -1 is the directrix, then the equation of the parabola is $$y = \frac{1}{4}x^2$$.

We value your feedback to improve our textbook solutions.

## Access millions of textbook solutions in one place

• Access over 3 million high quality textbook solutions
• Access our popular flashcard, quiz, mock-exam and notes features

## Join over 22 million students in learning with our Vaia App

The first learning app that truly has everything you need to ace your exams in one place.

• Flashcards & Quizzes
• AI Study Assistant
• Smart Note-Taking
• Mock-Exams
• Study Planner