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 .
Compare the given equation with the standard form x 2 = 4 a y to find 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 its vertex at the origin and opening upwards is x 2 = 4 a y , where ′ a ′ is the distance from the vertex to the focus and from the vertex to the directrix.
Finding the Value of 'a' Comparing the given equation x 2 = 2 y with the standard form x 2 = 4 a y , we can write 4 a = 2 . Solving for a , we get a = 4 2 = 2 1 .
Finding the Focus Since the parabola opens upwards and its vertex is at the origin ( 0 , 0 ) , the focus is at ( 0 , a ) , which is ( 0 , 2 1 ) .
Finding the Directrix The directrix is a horizontal line y = − a , so the equation of the directrix is 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, such as physics and engineering. For instance, the trajectory of a projectile under gravity (ignoring air resistance) follows a parabolic path. Knowing the focus and directrix of this parabola can help in determining the optimal launch angle and range of the projectile. Similarly, parabolic reflectors are used in satellite dishes and car headlights to focus signals or light at a single point (the focus), maximizing efficiency.
The focus of the parabola defined by x 2 = 2 y is located at ( 0 , 2 1 ) , and the equation of the directrix is y = − 2 1 . Therefore, the correct choice is B.
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