Note that to complete the square for x we added not
Here we have an ellipse with major axis parallel to the y axis since the bigger denominator is under the y term.
==>
==>
==>
Center:
vertices:
and
foci:
and
39+. Find the equation of the ellipse with center (3, 4), major axis of length 4, foci (3, 3) and (3, 5)
Solution:
Solution:
The bigger number (25) is under the . This tell us that the major axis of the ellipse is parallel to the y-axis (a vertical ellipse).
==>
==>
==>
Therefore,
vertices:
and
foci:
and
23. Find the vertices and foci of the ellipse
Solution:
We need to complete the square for both variable.
Regroup and move the constant to the other side.
Pull out the coefficient of and leave room for completing the square.
Square both sides
Multiply both sides by
Divide by 81
This is the equation of an ellipse with major axis parallel to the y axis.
center;
vertices:
and
foci:
and
We now know that the given equation is the equation of half an ellipse. Which half? From the equation we can see that all the x values will be greater than -5. Thus, the equation represents the right side of the ellipse with the above properties.
major axis of length 4 ==>
The equation is:
49. Find the equation of the ellipse with eccentricity 2/5, foci at (-1, 3) and (3, 3).
Solution:
==>
==>
(1)
The center of the ellipse will be at the middle point between the two foci.
c is half the distance between the two foci. ==>
Substitute c in equation (1), we get
Thus, the equation of the ellipse is
E1. Sketch the graph of
Solution:
==>
Rearrange the equation to make the quadratic variable on one side and the other variable on the other side.
Take 2 as a common factor and leave room for completing the square.
Factor out the coefficient of y
Divide by 2
Thus,
This is a equation of a parabola that opens upward, with:
and
vertex
focus;
directrix:
axis of symmetry:
Solution of Homework Problems Minus One

Chapter 8
8.1
10+. Find the vertex, focus and directrix of the parabola
Solution:
The equation is not in the standard form . Thus, we must first right it in the standard form.
Take 2 as a common factor
Use the property
Divide the equation by 4
This is the standard form of a parabola that opens to the left.
vertex:
focus:
directrix:
21+. Find the vertex, focus and directrix of the parabola
Solution:
Multiply the equation by -1 to make the squared term positive and rearrange the equation
==>
36+. The LNB (in instrument to collect satellite signals) is to be placed at the focus of a paraboloid dish. If the dish has diameter 120 cm and a depth of 40 cm, how far from the vertex of the paraboloid should the LNB be place.
Solution:
This problem was discussed in class.
E1. Sketch the graph of
Solution:
==>
This is the equation of a parabola that opens to the left.
==>
vertex:
focus:
directrix:
Axis of symm:
Thus, the given equation represents half a parabola. The question is: "Is it the upper half or the lower half?" A closer look at the given equation tells us that the y values will always be greater than 2. This helps us choose the upper half of the parabola. The graph below illustrates the two halves of the parabola.
8.2
1. Find the vertices and foci of the ellipse
31. Find the equation in standard form of the parabola with focus (3, -3) and directrix y = -5 .
Solution:
The equation of the directrix y = constant ==> the parabola opens either upward or downward. Moreover, the directrix is below the focus (by comparing the values -5 and -3) tells us that the parabola opens upward.
Therefore, the equation of the parabola will have the form
Now, the point on the directrix directly below the focus will have coordinates (3, -5). We can now use the midpoint formula to find the vertex, which is in the middle between the point (3, -5) and the focus (3, -3)
To find p we have to choices:

1) Compute the distance between the vertex and the focus
2) Compute half the distance between the focus and the point (3, -5) on the directrix.
Thus, the equation of the parabola is given by:
33. Find the equation in standard form of the parabola that has vertex (-4 , 1), has its axis of symmetry parallel to the y-axis, and passes through the point (-2, 2)
Solution:
A parabola with axis of symmetry parallel to the y-axis has the form:
Substitute the vertex
The parabola passes through the point (-2, 2) ==> the point satisfies the equation of the parabola.
Substitute the point in the equation to compute p
==>
The equation is: