College Algebra Questions and Problems With Solutions
sample 9 : Equation of Parabola
College algebra questions and problems on the equation of parabolas are presented. The solutions with explanations are located at the bottom of the page.
Question 1
Find an equation of the parabola with focus at (0 , 4) and vertex at (0 , 0).Question 2
Find an equation of the parabola with vertex at (0 , 0), the x axis is its axis of symmetry and its graph contains the point (-2 , 4).Question 3
Find an equation of the parabola with vertex at (0 , 2) and focus at (0 , 6).Question 4
Find the vertex, the focus, the axis of symmetry and the directrix of the parabola defined by the equationQuestion 5
Find the vertex, the focus, the axis of symmetry and the directrix of the parabola defined by the equationQuestion 6
Find an equation of the parabola with vertex at (-2 , -2) and focus at (-2 , -8).Question 7
Write an equation for the parabola shown in the graph below and find the focus of the parabola..
Question 8
Find an equation of the parabola with focus at (8 , 0) and directrix given by the equation x = 2.Question 9
Find an equation of the parabola with directrix given by the equation y = 2, a focus on the y axis, and the point (-6 , -8) lies on the parabola.Question 10
A parabolic dish with a diameter of 200 cm and a maximum depth of 50 cm is shown below. Find the focus of the dish.
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Solutions to the Above Questions
Solution Question 1
The distance from the vertex (0 , 0) to the focus (0 , 4) is |a| = 4. Since the vertex is at (0 , 0) and the focus is at (0 , 4) on the y axis, the parabola opens upward which means a = 4 and its equation is given bySolution Question 2
Since the x axis is the axis of symmetry of the parabola and its vertex is at the origin, the equation of the parabola has the formThe point (-2 , 4) lies on the parabola: 4 2 = 4a (-2)
Solve for a: a = -2 ; the equation is: y 2 = -8 x
Solution Question 3
Find an equation of the parabola with vertex at (0 , 2) and focus at (0 , 6).Since the vertex is at (0 , 2) and the focus is at (0 , 6), the parabola opens upward and the equation of a parabola with vertex at (h , k) is given by
h = 0 and k = 2; the equation is given by: x 2 = 4a (y - 2)
|a| = distance from vertex to focus = 4 , since parabola opens upward a = 4
the equation is given by: x 2 = 16 (y - 2)
Solution Question 4
We first complete the square using the terms in y and y 2 and write the given equation in the form (y - k) 2 = 4a (x - h) where (h , k) is the vertex and the focus is at (h + a , k), the axis of symmetry is given by y = k and the directrix is given by x = h - a.2(y 2 + 4y) + x + 1 = 0
2((y + 2) 2 - 4) + x + 1 = 0
(y + 2) 2 = - (1/2)(x - 7)
vertex at (7 , -2)
-(1/2) = 4a , hence a = -1/8
focus at (7 - 1/8 , -2) = (6.875 , -2)
axis of symmetry given by y = -2
Directrix is a vertical line given by: x = h - a = 7 + 1/8 = 7.125
Solution Question 5
We first complete the square using the terms in x and x 2 and write the given equation in the form (x - h) 2 = 4a (y - k) where (h , k) is the vertex and the focus is at (h , k + a), the axis of symmetry is given by x = h and the directrix is given by y = k - ax 2 - 8x - y + 2 = 0
((x - 4) 2 - 16) - y + 2 = 0
(x - 4) 2 = (y + 14)
vertex at (4 , -14)
1 = 4a , hence a = 1/4
focus at (4 , -14 + 1/4) = (4 , -13.75)
axis of symmetry given by x = 4
Directrix is a horizontal line given by: y = k - a = - 14 - 1/4 = -14.25
Solution Question 6
The vertex and the focus are on the same vertical line x = -2 with the focus below the vertex, therefore the parabola opens downward and its equation has the form(x - h) 2 = 4a (y - k) , vertex at (h , k)
with the vertex at (h , k) = (-2, -2), the equation is give by
(x + 2) 2 = 4a (y + 2)
The distance from the vertex (-2 , -2) to the focus (-2 , -8) is |a| = 6. Since the parabola opens downward a = -6.
The equation of the parabola is given by: (x + 2) 2 = -24 (y + 2)
Solution Question 7
The vertex is at (2 , 4)(y - k) 2 = 4a (x - h) , vertex at (h , k)
with the vertex at (h , k) = (2, 4), the equation is give by
(y - 4) 2 = 4a (x - 2)
Use the point (-2 , -2) to find a as follows
(-2 - 4) 2 = 4a (-2 - 2)
solve the above for a: a = -2.25
Equation of the parabola: (y - 4) 2 = -9 (x - 2)
The focus is at (h + a , k) = (2 - 2.25 , 4) = (-0.25 , 4)
Solution Question 8
Since the directrix is the x = 2 and the focus is at (8 , 0), the parabola has the x axis as an axis of symmetry and opens to the right. Its equation is of the form(y - k) 2 = 4a (x - h) , vertex at (h , k)
The vertex is the midpoint of the point (2 , 0), which the point of intersection of the directrix and the x axis, and the focus ( 8 , 0). Hence h = 5 and k = 0. The distance between the directrix and the focus is twice |a|. Hence 2|a| = 6, |a| = 3 and since the parabola opens to the right a = 3.
y 2 = 12 (x - 5)
Solution Question 9
The focus is on the y axis and therefore has the form F(0 , b). The distance from the given point to the directrix is equal to 10. By definition any point on the parabola, and particular point (- 6 , -8), should be at equal distances from the directrix and the focus. Hence10 = √(6 2 + (b + 8)) 2 )
Solve the above for b: 2 solutions b = 0 and b = -16
THERE ARE TWO SOLUTIONS TO THIS PROBLEM
First solution: the distance from the focus F(0 , 0) to the directrix y = 2 is equal to 2|a|, hence |a| = 1 and a = -1 since parabola opens downward. The vertex is at (0 , 1), at equal distances from the focus and the directrix.
equation: x 2 = - 4 (y - 1)
Second solution: the distance from the focus F(0 , -16) to the directrix y = 2 is equal to 2|a|, hence |a| = 9 and a = -9 since parabola opens downward. The vertex is at (0 , -7), at equal distances from the focus and the directrix.
equation: x 2 = - 36 (y + 7)
Solution Question 10
The equation of the parabolic dish is of the form: x 2 = 4 a yPoint (100 , 50) lies on the graph of the parabolic dish, hence
100 2 = 4 a * 50
solve to find: a = 50 which is also the distance from the vertex at (0 , 0) to the focus. Hence the focus is at (0 , 50 cm).
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