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 the parabola as the set of points equidistant from the focus ( 8 , 0 ) and the directrix x = − 8 .
Express the distance from a point ( x , y ) on the parabola to the focus and to the directrix.
Equate the two distances and simplify the resulting equation by squaring, expanding, and canceling terms.
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 .
Definition of a Parabola The definition of a parabola is the set of all points ( x , y ) that are equidistant to the focus and the directrix. We need to use this definition to derive the equation.
Distance to the Focus The distance from a point ( x , y ) to the focus ( 8 , 0 ) is given by the distance formula: d 1 = ( x − 8 ) 2 + ( y − 0 ) 2 = ( x − 8 ) 2 + y 2
Distance to the Directrix The distance from a point ( x , y ) to the directrix x = − 8 is the perpendicular distance to the line, which is given by: d 2 = ∣ x − ( − 8 ) ∣ = ∣ x + 8∣
Equating the Distances Since the point ( x , y ) lies on the parabola, the distances d 1 and d 2 must be equal: ( x − 8 ) 2 + y 2 = ∣ x + 8∣
Squaring Both Sides To eliminate the square root, square both sides of the equation: ( x − 8 ) 2 + y 2 = ( x + 8 ) 2
Expanding the Equation Expand both sides of the equation: x 2 − 16 x + 64 + y 2 = x 2 + 16 x + 64
Simplifying the Equation Simplify the equation by canceling out the x 2 and 64 terms: y 2 − 16 x = 16 x
Isolating the y^2 Term Isolate the y 2 term: 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 design of satellite dishes and reflective telescopes. The parabolic shape allows incoming signals or light to be focused at a single point, improving signal strength or image clarity. Understanding the equation of a parabola helps engineers design these systems effectively, ensuring optimal performance in communication and astronomical observation.
