A parabola is represented by the equation $x^2=4 y$. What are the coordinates of the focus of the parabola?
A. $(0,4)$
B. $(0,1)$
C. $(1,0)$
D. $(4,0)$
Answer (1)
The equation of the parabola is x 2 = 4 y .
The general form of a parabola is x 2 = 4 a y , where the focus is at ( 0 , a ) .
Comparing the given equation with the general form, we find 4 a = 4 , so a = 1 .
The coordinates of the focus are ( 0 , 1 ) .
Explanation
Problem Analysis We are given the equation of a parabola as x 2 = 4 y . Our goal is to find the coordinates of the focus of this parabola.
General Form of a Parabola The general equation of a parabola that opens upwards with its vertex at the origin is given by x 2 = 4 a y , where ′ a ′ is the distance from the vertex to the focus. The coordinates of the focus are ( 0 , a ) .
Comparing Equations Comparing the given equation x 2 = 4 y with the general form x 2 = 4 a y , we can see that 4 a = 4 .
Solving for a To find the value of a , we solve the equation 4 a = 4 :
4 a = 4 a = 4 4 a = 1
Finding the Focus Since the focus of the parabola is at ( 0 , a ) , and we found that a = 1 , the coordinates of the focus are ( 0 , 1 ) .
Final Answer Therefore, the coordinates of the focus of the parabola x 2 = 4 y are ( 0 , 1 ) .
Examples
Understanding parabolas is crucial in various fields, such as physics and engineering. For example, the trajectory of a projectile (like a ball thrown in the air) follows a parabolic path. Knowing the focus of this parabola can help in calculating the range and maximum height of the projectile. Similarly, parabolic reflectors are used in satellite dishes and car headlights to focus signals or light at a specific point (the focus), maximizing efficiency.
