12.7 Sample test: distance, circles, and parabolas

Below are 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?

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

Find all the points that are exactly 4 units away from the origin, where the $x$ and $y$ coordinates are the same .

Find the equation for a parabola whose vertex is $\mathrm{(3,-1)}$ , and that contains the point $\mathrm{(0,4)}$ .

$2x^2+2y^2–6x+4y+2=0$

• 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!)

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

• 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.

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

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

In the drawing above, I show the focus and the directrix, and an arbitrary point ( $x$ , $y$ ) on the parabola.

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