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Question 308473: identify the conic defined by each equation. If it is a parabola, give its vertex, focus, and directrix; if an ellipse, give its center, vertices, and foci; if a hyperbola, give its center, vertices, foci and asymptotes. Draw a graph with these features identified.
8y = (x −1)^2 − 4
I think this is a parabola I am not sure. Once I know what type of conic the equations are then I know how to solve then I just have trouble determining the difference between the three.
My second question is this second equation. I think this one is an ellipse but I just want to be sure.
2x^2 + 3y^2 + 4x − 6y =13
My third question is this third equation which I think is an hyperbola.
(x+1)^2/4-y^2/9=1
My fourth question of the day is....
A parabolic reflector (paraboloid of revolution) is used by TV crews of a football game to pick up
the referee’s announcements, quarterback signals, and so on. A microphone is placed at the focus
of the parabola. If a certain reflector is 4 feet wide and 1.5 feet deep, where would the microphone
be placed? Include a well labeled sketch with your answer.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
If there is a squared x term and not a squared y term (or vice versa) then you have a parabola. If x is the squared term and there is no xy term, then the axis is a vertical line. If y is the squared term and there is no xy term, then the axis is a horizontal line. If there is an xy term, then the axis is an oblique line.
If you have both an x squared term and a y squared term and the coefficients on these two terms are equal, then you have a circle.
If the coefficients are unequal, but the signs are the same, then you have an ellipse. The term with the smaller coefficient (or larger denominator on the coefficient) identifies the major axis. Unless there is an xy term in which case the axes of the ellipse are oblique to the coordinate axes.
If the coefficients are unequal and the signs are different, then you have a hyperbola.
So, all three of your "I thinks" are correct.
For your last one, plot your parabola on a graph. Plot a point at (-2,0), another at (2,0), and a third one at (0,-1.5).
Now we need the equation of a parabola that passes through all three points.
Is the general equation of a parabola. But we know that the parabola we want must pass through (-2,0), so:
^2\ +\ b(-2)\ +\ c\ =\ 0)
and it must pass through (2, 0), so:
^2\ +\ b(2)\ +\ c\ =\ 0)
and it must pass through (0, -1.5), so:
^2\ +\ b(0)\ +\ c\ =\ -1.5)
Which we can rewrite as the system of equations:
Ah, ha...
So now we have:
Add the two equations:
And we have 
I'll leave it to you to verify that 
So, let's write our equation:
Next, let's put it into ^2\ =\ 4p(y\ -\ k)) form:
)
So,
Which is to say
And that means our focus is  ft distant from the vertex which we established at (0,-1.5).
So:
And the focus is at ) , which is 8 inches (2/3 foot) along the axis from the lowest point of the dish reflector.
John
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