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Chapter 7: Problem 30

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: \((7,-1) ;\) Directrix: \(y=-9\)

Short Answer

Expert verified
The standard form of the equation of the parabola is \(x = 4(y + 5)^2 + 7\)

Step by step solution

01

Calculate the vertex

The vertex of the parabola is midway between the focus and the directrix. Since the directrix is \(y = -9\) and the y-coordinate of the focus is -1, the y-coordinate of the vertex is the average of these two values. So, \(y = (-9 + (-1))/2 = -5\). The x-coordinate of the vertex is the same as the x-coordinate of the focus, which is 7. So the vertex is \( (7, -5) \)
02

Calculate the value of p

The value of p is the distance from the vertex to the focus. In this case, since the vertex and the focus have the same x-coordinate, p is simply the difference in their y-coordinates, which is \( -1 - (-5) = 4 \)
03

Write the equation of the parabola

The general form of the equation of a parabola with a horizontal axis is \(x = py^2 + h\). We found earlier that p = 4 and h = 7, so the standard form of the equation of the parabola is \(x = 4(y + 5)^2 + 7 \)

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Most popular questions from this chapter

Use a graphing utility to graph the parabolas.Write the given equation as a quadratic equation in \(y\) and use the quadratic formula to solve for \(y .\) Enter each of the equations to produce the complete graph. $$y^{2}+10 y-x+25=0$$

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$4 y^{2}=1-4 x^{2}$$

If you are given the standard form of the equation of a parabola with vertex at the origin, explain how to determine if the parabola opens to the right, left, upward, or downward.

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$7 x^{2}=35-5 y^{2}$$

Use a graphing utility to graph \(\frac{x^{2}}{4}-\frac{y^{2}}{9}=0 .\) Is the graph a hyperbola? In general, what is the graph of \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=0 ?\)
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