A. Determine the focus and directrix of the parabola with the given standard equation.
Sketch the graph and indicate the focus, directrix, vertex and axis of symmetry.
1. x2
=36y 4. (y+1)2
=20(x+3)
2. x
2
=-25y 5. x2
-6x-4y+5=0
3. y
2
=24(x+1) 6. y2
+6x+2y-17=0
B. Find the standard equation of a parabola which satisfies the given conditions.
1. vertex V(2,-3), focus F(2,8)
2. vertex V(-5,-1), focus F(10,-1)
3. vertex V(6,-4), directrix x=4
4. vertex V(2,-5), vertical axis, through (8,-3)
5. vertex V(1,-7), horizontal axis, through (5,2)
Sketch the graph and indicate the focus, directrix, vertex and axis of symmetry.
1. x2
=36y 4. (y+1)2
=20(x+3)
2. x
2
=-25y 5. x2
-6x-4y+5=0
3. y
2
=24(x+1) 6. y2
+6x+2y-17=0
B. Find the standard equation of a parabola which satisfies the given conditions.
1. vertex V(2,-3), focus F(2,8)
2. vertex V(-5,-1), focus F(10,-1)
3. vertex V(6,-4), directrix x=4
4. vertex V(2,-5), vertical axis, through (8,-3)
5. vertex V(1,-7), horizontal axis, through (5,2)
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Determine the focus and directrix of the parabola with the given standard equation. Sketch the graph and indicate the focus, directrix, vertex and axis of symmetry.
Problem 1:
1.) x² = 36y
x² = 4ay
(x - 0)² = 4(9)(y - 0)
x² = 36y
4a = 36
a = 9
h = 0 ; k = 0 ; a = 9
Vertex (h , k)
Vertex (0 , 0)
Focus (0, a)
Focus (0, 9)
Directrix: y = -a
Directrix: y = -9
Axis of symmetry: x = 0
Problem 2:
2.) x² = -25y
x² = -4ay
(x - 0)² = -4(25/4)(y - 0)
x² = -25y
4a = -25
a = -25/4
h = 0 ; k = 0 ; a = -25/4
Vertex (h , k)
Vertex (0 , 0)
Focus (0, a)
Focus (0, -25/4)
Directrix: y = -a
Directrix: y = -(-25/4)
Directrix: y = 25/4
Axis of symmetry: x = 0
Problem 3:
3.) y² = 24(x + 1)
(y - 0)² = 24(x + 1)
(y - 0)² = 4(6)(x + 1)
y² = 4ax
y² = 24x
4a = 24
a = 24/4
a = 6
h = -1 ; k = 0 ; a = 6
Vertex (h , k)
Vertex (-1 , 0)
Focus(a , 0)
X = x + 1 ; Y = y - 0
x = X - 1 ; y = Y + 0
x = 6 - 1 = 5 ; y = 0 + 0 = 0
Focus (5, 0)
Directrix: X = -a
X = x + 1
x = X - 1
x = -6 - 1 = -7
Directrix: x = -7
Y = y - 0
y = Y + 0
y = 0 + 0
Axis of symmetry: y = 0
Problem 4:
4.) (y + 1)² = 20(x + 3)
(y + 1)² = 20(x + 3)
(y + 1)² = 4(5)(x + 3)
y² = 4ax
y² = 20x
4a = 20
a = 20/4
a = 5
h = -3 ; k = -1 ; a = 5
Vertex (h , k)
Vertex (-3 , -1)
Focus(a , 0)
X = x + 3 ; Y = y + 1
x = X - 3 ; y = Y - 1
x = 5 - 3 = 2 ; y = 0 - 1 = -1
Focus (2, -1)
Directrix: X = -a
X = x + 3
x = X - 3
x = -5 - 3 = -8
Directrix: x = -8
Y = y + 1
y = Y - 1
y = 0 - 1
Axis of symmetry: y = -1
Problem 5:
5.) x² - 6x - 4y + 5 = 0
x² - 6x = 4y - 5
x² - 6x + 3² = 4y - 5 + 3²
(x - 3)² = 4y - 5 + 9
(x - 3)² = 4y + 4
(x - 3)² = 4(y + 1)
x² = 4ay
4a = 4
a = 4/4
a = 1
h = 3 ; k = -1 ; a = 1
Vertex (h , k)
Vertex (3 , -1)
Focus(0 , a)
X = x - 3 ; Y = y + 1
x = X + 3 ; y = Y - 1
x = 0 + 3 = 3 ; y = 1 - 1 = 0
Focus (3, 0)
Directrix: Y = -a
Y = y + 1
y = Y - 1
y = -1 - 1 = -2
Directrix: y = -2
X = x - 3
x = X + 3
x = 0 + 3
Axis of symmetry: x = 3
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