The parabola has a focus at $(-3,0)$ and directrix $x=3$. What is the correct equation for the parabola?
A. $x^2=-12 y$
B. $x^2=3 y$
C. $y^2=3 x$
D. $y^2=-12 x$
Answer (1)
Define the distances from a point (x, y) on the parabola to the focus and the directrix.
Equate these distances based on the definition of a parabola.
Simplify the resulting equation by squaring, expanding, and canceling terms.
Identify the equation of the parabola: y 2 = − 12 x .
Explanation
Data and problem analysis The focus of the parabola is at the point ( − 3 , 0 ) .
Data and problem analysis The directrix of the parabola is the vertical line x = 3 .
Objective The definition of a parabola states that for any point ( x , y ) on the parabola, the distance to the focus is equal to the distance to the directrix.
Solution plan Let ( x , y ) be a point on the parabola. The distance from ( x , y ) to the focus ( − 3 , 0 ) is given by the distance formula: ( x − ( − 3 ) ) 2 + ( y − 0 ) 2 = ( x + 3 ) 2 + y 2 The distance from ( x , y ) to the directrix x = 3 is the perpendicular distance to the line, which is ∣ x − 3∣ .
Equating the distances Setting the distance to the focus equal to the distance to the directrix, we have: ( x + 3 ) 2 + y 2 = ∣ x − 3∣
Squaring both sides Squaring both sides of the equation to eliminate the square root and absolute value: ( x + 3 ) 2 + y 2 = ( x − 3 ) 2
Expanding the equation Expanding both sides: x 2 + 6 x + 9 + y 2 = x 2 − 6 x + 9
Simplifying Simplifying the equation by canceling out the x 2 and 9 terms: 6 x + y 2 = − 6 x
Isolating y^2 Isolating the y 2 term: y 2 = − 12 x
Finding the correct equation Comparing the derived equation with the given options, we find that the correct equation for the parabola is y 2 = − 12 x .
Final Answer The equation of the parabola is y 2 = − 12 x .
Examples
Parabolas are commonly found in the design of satellite dishes and reflecting telescopes. The parabolic shape focuses incoming signals or light to a single point (the focus), allowing for efficient signal reception or image formation. Understanding the equation of a parabola helps engineers design these devices to optimize their performance.
