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Chapter 7: Problem 26

Find the standard form of the equation of each parabola satisfying the given conditions. Vertex: \((5,-2) ;\) Focus: \((7,-2)\)

Short Answer

Expert verified
The standard form equation of the parabola with vertex \((5,-2)\) and focus \((7,-2)\) is \((y+2)^2 = 8(x-5)\).

Step by step solution

01

Determine the direction of the parabola

In this step we figure out the direction of the parabola. The vertex of the parabola is \((5,-2)\) and the focus is \((7,-2)\). Noting the coordinates of the vertex and focus, the focus lies to the right of the vertex, hence the parabola opens to the right.
02

Calculate the value of \(p\)

The value of \(p\) is the distance from the vertex to the focus point. We calculate the distance using their x-coordinates since the y-coordinates are the same, which simplifies the distance to the difference of the x-coordinates. The distance \(p = 7 - 5 = 2\).
03

Write the equation in standard form

Now that we have the value of \(p\), we can fill the values into the standard form equation of a parabola. Our vertex is \((h, k) = (5,-2)\) and \(p = 2\). Substituting these into the standard form equation for a parabola that opens to the right gives: \((y-k)^2 = 4p(x-h)\) which simplifies to \((y+ 2)^2 = 8(x - 5)\).

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Most popular questions from this chapter

Convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. $$4 x^{2}-9 y^{2}-16 x+54 y-101=0$$

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$6 x^{2}=30-5 y^{2}$$

In Exercises 43-50, convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. $$x^{2}-y^{2}-2 x-4 y-4=0$$

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$4 x^{2}+16 y^{2}=64$$

Graph \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) and \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1\) in the same viewing rectangle for values of \(a^{2}\) and \(b^{2}\) of your choice. Describe the relationship between the two graphs.
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