Write an equation of a parabola that opens to the left, has a vertex at the origin, and a focus at ( [tex]$(-9,0)$[/tex] ).
Answer (1)
The parabola opens to the left, so its equation is of the form x = a y 2 .
The focus is at ( − 9 , 0 ) , which means 4 a 1 = − 9 .
Solving for a , we find a = − 36 1 .
Therefore, the equation of the parabola is x = − 36 1 y 2 .
Explanation
Understanding the Problem We are given that the parabola opens to the left, has a vertex at the origin, and a focus at ( − 9 , 0 ) . We need to find the equation of this parabola.
General Equation The general equation of a parabola that opens to the left with a vertex at the origin is given by x = a y 2 , where a < 0 . The focus of such a parabola is at ( 4 a 1 , 0 ) .
Using the Focus Information We are given that the focus is at ( − 9 , 0 ) . Therefore, we have 4 a 1 = − 9 .
Solving for a Solving for a , we get 1 = − 36 a , which implies a = − 36 1 .
Finding the Equation Substituting this value of a into the general equation x = a y 2 , we get x = − 36 1 y 2 .
Final Answer Therefore, the equation of the parabola is x = − 36 1 y 2 .
Examples
Parabolas are not just abstract mathematical concepts; they have real-world applications. For example, the reflective surface of a flashlight or a satellite dish is often shaped like a parabola. This shape allows the light or radio waves to be focused at a single point, making the signal stronger. Understanding the properties of parabolas helps engineers design these devices efficiently.
