A parabola can be represented by the equation $x^2=2 y$. What are the coordinates of the focus and the equation of the directrix?
A. focus: $(0,8)$; directrix: $y=-8$
B. focus: $\left(0, \frac{1}{2}\right)$; directrix: $y=-\frac{1}{2}$
C. focus: $(8,0)$; directrix: $x=-8$
D. focus: $\left(\frac{1}{2}, 0\right)$; directrix: $x=-\frac{1}{2}$
Answer (2)
The equation of the parabola is given as x 2 = 2 y .
Rewrite the equation in the standard form x 2 = 4 a y and find the value of a = 2 1 .
The focus of the parabola is at ( 0 , 2 1 ) .
The equation of the directrix is y = − 2 1 .
focus: ( 0 , 2 1 ) ; directrix: y = − 2 1
Explanation
Problem Analysis We are given the equation of a parabola as x 2 = 2 y . Our goal is to find the coordinates of the focus and the equation of the directrix.
Standard Form of a Parabola The standard form of a parabola with a vertical axis of symmetry is x 2 = 4 a y , where the focus is at ( 0 , a ) and the directrix is y = − a . We need to rewrite the given equation in this standard form to identify the value of a .
Finding the Value of a Comparing the given equation x 2 = 2 y with the standard form x 2 = 4 a y , we can set 4 a = 2 . Solving for a , we get:
4 a = 2
a = 4 2
a = 2 1
Finding the Focus and Directrix Now that we have the value of a , we can find the coordinates of the focus and the equation of the directrix.
The focus is at ( 0 , a ) = ( 0 , 2 1 ) .
The equation of the directrix is y = − a = y = − 2 1 .
Final Answer Therefore, the coordinates of the focus are ( 0 , 2 1 ) and the equation of the directrix is y = − 2 1 .
Examples
Understanding parabolas is crucial in various fields like physics and engineering. For example, satellite dishes and reflectors in car headlights are designed with parabolic shapes. The focus of a parabola is the point where incoming parallel rays converge after reflection, which is why satellite dishes are shaped like parabolas to focus radio waves onto a receiver placed at the focus. Similarly, in architecture, parabolic arches provide excellent structural support, distributing weight evenly.
The coordinates of the focus for the parabola x 2 = 2 y are ( 0 , 2 1 ) and the equation of the directrix is y = − 2 1 . The correct answer is option B.
;
