   Chapter 10.5, Problem 29E

Chapter
Section
Textbook Problem

# Identify the type of conic section whose equation is given and find the vertices and foci.29. 3x2 − 6x − 2y = 1

To determine

To Find: The type of conic section, vertices, and foci for the equation 3x26x2y=1 .

Explanation

Given:

The equation is as follows.

3x26x2y=1 (1)

Rewrite the equation (1).

3x26x+33=2y+13x26x+3=2y+1+33(x22x+1)=2y+4

Factorize the above equation as below.

3(x1)2=2y+43(x1)2=2(y+2)

(x1)2=23(y+2) (2)

Then, compare the equation (2) with the standard equation of parabola.

x2=4py

Therefore, the type of conic section is parabola.

Calculation:

Compute the vertex of the parabola using the equation.

(xh)2=4p(yk)(x(1))2=4p(y(2))(h,k)=(1,2)

Therefore, the vertex of the parabola (h,k) is (1,2)

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