Which is the standard form of the equation of a parabola with a focus of $(8,0)$ and directrix $x=-8$?
A. $y^2=-8 x$
B. $y^2=8 x$
C. $y^2=32 x$
D. $y^2=-32 x$
Answer (1)
Define a parabola as the set of points equidistant from the focus and directrix.
Express the distance from a point ( x , y ) on the parabola to the focus ( 8 , 0 ) and the directrix x = − 8 .
Equate the two distances and simplify the resulting equation.
Obtain the standard form of the parabola equation: y 2 = 32 x .
Explanation
Problem Analysis The problem asks for the standard form of the equation of a parabola given its focus and directrix. The focus is at ( 8 , 0 ) and the directrix is the line x = − 8 . We need to find the equation of the parabola.
Set up the equation A parabola is defined as the set of all points equidistant to the focus and the directrix. Let ( x , y ) be a point on the parabola. The distance from ( x , y ) to the focus ( 8 , 0 ) is given by the distance formula: ( x − 8 ) 2 + ( y − 0 ) 2 The distance from ( x , y ) to the directrix x = − 8 is the perpendicular distance, which is: ∣ x − ( − 8 ) ∣ = ∣ x + 8∣ We set these two distances equal to each other:
Simplify the equation ( x − 8 ) 2 + y 2 = ∣ x + 8∣ To eliminate the square root, we square both sides of the equation: ( x − 8 ) 2 + y 2 = ( x + 8 ) 2 Expanding both sides, we get: x 2 − 16 x + 64 + y 2 = x 2 + 16 x + 64 Now, we simplify the equation by canceling out the x 2 and 64 terms: y 2 − 16 x = 16 x y 2 = 32 x
Final Answer The standard form of the equation of the parabola is y 2 = 32 x .
Examples
Parabolas are commonly found in the real world, such as the curve of a satellite dish or the trajectory of a projectile. Understanding how to find the equation of a parabola given its focus and directrix allows engineers to design these structures and predict their behavior. For example, knowing the focus and directrix of a satellite dish helps in positioning the receiver to maximize signal reception. Similarly, understanding the parabolic trajectory of a ball helps athletes to improve their throwing or kicking techniques.
