12 multiple-choice questions, progressively harder.
How far is PPP at (−6,4)(-6,4)(−6,4) from the directrix y=−3y=-3y=−3?
Solution
Correct answer: B
The directrix is horizontal, so use the vertical gap.
The x-coordinate does not affect this distance.
The focus is (2,7)(2,7)(2,7) and the directrix is y=1y=1y=1. Where is the vertex?
Correct answer: D
The foot of the focus is (2,1)(2,1)(2,1). The vertex is their midpoint.
The x-coordinate stays 2.
For (x−5)2=28(y+1)(x-5)^2=28(y+1)(x−5)2=28(y+1), what is the signed parameter ppp?
Correct answer: A
The multiplier is 4p4p4p.
Its positive sign means the focus is above the vertex.
What is the focus of x2=−24yx^2=-24yx2=−24y?
Correct answer: C
The vertex is the origin and the axis is vertical.
Move six units down from the vertex.
Which is the directrix of (y+3)2=16(x−2)(y+3)^2=16(x-2)(y+3)2=16(x−2)?
The vertex is (2,−3)(2,-3)(2,−3) and the axis is horizontal. Here p=4p=4p=4.
The directrix lies four units left of the vertex.
Which way does (y−6)2=−12(x+4)(y-6)^2=-12(x+4)(y−6)2=−12(x+4) open?
The y-coordinate is squared, so the axis is horizontal. The multiplier is negative.
The focus lies left of the vertex, so the curve opens left.
A parabola has focus (−1,5)(-1,5)(−1,5) and directrix y=−1y=-1y=−1. What is its axis?
The axis goes through the focus perpendicular to the horizontal directrix.
It is a vertical line.
Which distance equality describes focus (3,1)(3,1)(3,1) and directrix x=−2x=-2x=−2?
The focus distance uses differences from (3,1)(3,1)(3,1). The line is vertical, so its perpendicular distance uses only the x-gap.
This gives ∣x+2∣|x+2|∣x+2∣.
For (x+2)2=−10(y−4)(x+2)^2=-10(y-4)(x+2)2=−10(y−4), what is the vertex-focus distance?
The signed parameter is −10/4=−5/2-10/4=-5/2−10/4=−5/2. A geometric distance is its absolute value.
The minus sign controls direction.
Which point is on the locus with focus (0,1)(0,1)(0,1) and directrix y=−1y=-1y=−1?
For (2,1)(2,1)(2,1), both distances are 2.
The line distance is ∣1+1∣=2|1+1|=2∣1+1∣=2. The other focus/line distances are 1 and 2, 0 and 2, and 5\sqrt55 and 1.
Which focus and directrix satisfy the required condition for a parabola?
The focus must be off the directrix. The valid pair has a vertical gap of five units.
Each other focus lies on its stated line.
Point PPP is on a parabola with focus FFF and directrix ddd. Its perpendicular foot on ddd is HHH. If PF=9PF=9PF=9, what is PHPHPH?
A locus point is equally distant from the focus and the directrix. The perpendicular segment measures the distance to the line.
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