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Sample Test: Distance, Circles, and Parabolas

Module by: Kenny M. Felder. E-mail the author

Summary: This module provides a sample test covering distance, circles, and parabolas.

Exercise 1

Below are the points (–2,4) and (–5,–3).

A picture of the points (–2,4) and (–5,–3).

  • a. How far is it across from one to the other (the horizontal line in the drawing)?
  • b. How far is it down from one to the other (the vertical line in the drawing)?
  • c. How far are the two points from each other?
  • d. What is the midpoint of the diagonal line?

Exercise 2

What is the distance from the point (–1024,3) to the line y = -1 y=-1?

Exercise 3

Find all the points that are exactly 4 units away from the origin, where the xx and yy coordinates are the same.

Exercise 4

Find the equation for a parabola whose vertex is (3,−1)(3,−1), and that contains the point (0,4)(0,4).

Exercise 5

2 x 2 + 2 y 2 – 6 x + 4 y + 2 = 0 2 x 2 +2 y 2 –6x+4y+2=0

  • a: Put this equation in the standard form for a circle.
  • b: What is the center?
  • c: What is the radius?
  • d: Graph it on the graph paper.
  • e: Find one point on your graph, and test it in the original equation. (No credit unless I can see your work!)

Exercise 6

x = - 1 4 y 2 + y + 2 x=- 1 4 y 2 +y+2

  • a: Put this equation in the standard form for a parabola.
  • b: What direction does it open in?
  • c: What is the vertex?
  • d: Graph it on the graph paper.

Exercise 7

Find the equation for a circle whose diameter stretches from (2,2) to (5,6).

Exercise 8

We’re going to find the equation of a parabola whose focus is (3,2) and whose directrix is the line x = -3x=-3. But we’re going to do it straight from the definition of a parabola.

The equation of a parabola whose focus is (3,2) and whose directrix is the line x=–3.

In the drawing above, I show the focus and the directrix, and an arbitrary point (xx,yy) on the parabola.

  • a: d 1 d1 is the distance from the point (xx,yy) to the focus (3,2). What is d 1 d1?
  • b: d 2 d2 is the distance from the point (xx,yy) to the directrix ( x = -3x=-3). What is d 2 d2?
  • c: What defines the parabola as such—what makes (xx,yy) part of the parabola—is that d 1 = d 2 d1=d2. Write the equation for the parabola.
  • d: Simplify your answer to part (c); that is, rewrite the equation in the standard form.

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