12 multiple-choice questions, progressively harder.
For 2x2−12x+16y+42=02x^2-12x+16y+42=02x2−12x+16y+42=0, which pair is the focus and directrix?
Solution
Correct answer: D
Divide by 2 and complete the square.
The vertex is (3,−3/2)(3,-3/2)(3,−3/2) and the signed parameter is −2-2−2. The focus is (3,−7/2)(3,-7/2)(3,−7/2) and the directrix is y=1/2y=1/2y=1/2.
A parabola with vertical axis has vertex (3,−2)(3,-2)(3,−2) and passes through (7,0)(7,0)(7,0). What is its focus?
Correct answer: B
Substitute the point into (x−3)2=4p(y+2)(x-3)^2=4p(y+2)(x−3)2=4p(y+2).
Move two units up from the vertex to locate the focus.
A left-opening parabola has vertex (5,2)(5,2)(5,2) and focus-directrix spacing 999. Which equation describes it?
Correct answer: C
Half the focus-directrix spacing is the vertex-focus distance. The left opening sets the sign.
The focus is (2,3)(2,3)(2,3) and directrix is x=−4x=-4x=−4. Which point lies on the locus?
Correct answer: A
At (2,9)(2,9)(2,9), the focus distance is 6 and the line distance is 6.
The other focus/line distances are 3 and 6, 3 and 9, and 13\sqrt{13}13 and 3.
A parabola has directrix y=−4y=-4y=−4 and focus (1,f)(1,f)(1,f). The point (1,2)(1,2)(1,2) is its vertex. What is fff?
The vertex is halfway between the directrix foot (1,−4)(1,-4)(1,−4) and the focus.
The vertex-focus and vertex-line distances are both 6.
The parabola (y−2)2=12(x+3)(y-2)^2=12(x+3)(y−2)2=12(x+3) moves 444 units right and 555 units down. What is the new directrix?
Originally the vertex is (−3,2)(-3,2)(−3,2) and the signed parameter is 333. The original directrix is x=−6x=-6x=−6. A move four units right changes its x-coordinate.
The vertical move does not change a vertical line's equation.
For (x+1)2=c(y−3)(x+1)^2=c(y-3)(x+1)2=c(y−3), the focus is (−1,1)(-1,1)(−1,1). What is ccc?
The vertex is (−1,3)(-1,3)(−1,3), so the signed vertical step to the focus is −2-2−2.
A right-opening parabola has directrix x=−5x=-5x=−5 and passes through (3,1)(3,1)(3,1). Its focus is (f,1)(f,1)(f,1). Which values of fff are possible?
The line distance is 8, so the horizontal focus distance is 8.
The value −5-5−5 places the focus on the directrix and is excluded. The remaining focus is to its right.
For (y+4)2=−4(x−6)(y+4)^2=-4(x-6)(y+4)2=−4(x−6), how far is the vertex from the directrix?
The signed parameter is −1-1−1. The vertex-to-directrix distance equals its distance to the focus.
A vertical-axis parabola has focus (a,4)(a,4)(a,4) and directrix y=−2y=-2y=−2. It passes through (5,4)(5,4)(5,4). Which values of aaa are possible?
The point is at the focus height. Its focus distance is therefore ∣5−a∣|5-a|∣5−a∣, and its line distance is 6.
Both focus positions remain off the directrix.
For x2+10x−12y+1=0x^2+10x-12y+1=0x2+10x−12y+1=0, which is the vertex-focus distance?
Complete the square.
Thus 4p=124p=124p=12.
The signed parameter is 333. The required distance is ∣p∣|p|∣p∣.
A parabola is (y−5)2=m(x+2)(y-5)^2=m(x+2)(y−5)2=m(x+2).
Its directrix is x=1x=1x=1. Which pair gives mmm and the focus?
The vertex is (−2,5)(-2,5)(−2,5). The directrix is three units right, so the focus is three units left.
The focus is (−5,5)(-5,5)(−5,5).
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