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Chapter 6: Problem 43

Find the vertex, focus, and directrix of the parabola. Use a graphing utility to graph the parabola. $$x^{2}+4 x+6 y-2=0$$

Short Answer

Expert verified
The vertex is (2, 0), the focus is (2, -1.5), and the equation of the directrix is y = 1.5.

Step by step solution

01

Rewrite Parabola Equation

Rewrite the given equation in the standard form of a parabola. That is, \( y = ax^2 + bx + c \). In our case, the equation can be written as \( y = -\frac{1}{6}x^2 - \frac{2}{3}x +\frac{1}{3} \).
02

Find the Vertex

The vertex of a parabola with equation \( y = ax^2 + bx + c \) has the coordinates h = -\frac{b}{2a} and k = c - \frac{b^2}{4a}. From the equation \( y = -\frac{1}{6}x^2 - \frac{2}{3}x +\frac{1}{3} \), the vertex (h, k) are h = -\frac{b}{2a} = -\frac{-2/3}{-1/3} = 2, and k = c - \frac{b^2}{4a} = \frac{1}{3} - \frac{(-2/3)^2}{4*(-1/6)} = 0. Therefore, the vertex is (2, 0)
03

find the Focus

The focus has the coordinates \( (h, k+ \frac{1}{4a} ) \). Using the values from step 2, it is (2, 0 + \frac{1}{4*(-1/6)}) = (2, -1.5)
04

Find the Directrix

The equation of the directrix is \( y = k - \frac{1}{4a} \). Substituting the value from step 2, it is y = 0 - \frac{1}{4*(-1/6)} = y = 1.5
05

Draw Parabola

To draw the parabola, plot the vertex, focus, and directrix from the previous steps on a graphing utility. These points will aid in drawing a smooth curve.

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Most popular questions from this chapter

In Exercises \(71-90,\) convert the rectangular equation to polar form. Assume \(a > 0\). $$3 x+5 y-2=0$$

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