Find the coordinates of the focus and equation of the directrix for the parabola given by $y^2=-4 x$.
The general formula for this parabola is $y^2=4 p x$.
Therefore, the value of $p$ is $\square$ .
The coordinates of the focus are $\square$ .
The equation of the directrix is $\square$ .
Answer (1)
We are given the equation of the parabola y 2 = − 4 x and the general form y 2 = 4 p x .
By comparing the given equation with the general form, we find that 4 p = − 4 , so p = − 1 .
The focus of the parabola is at ( p , 0 ) , which is ( − 1 , 0 ) .
The equation of the directrix is x = − p , which is x = 1 . Therefore, the final answer is p = − 1 , focus = ( − 1 , 0 ) , directrix = x = 1 .
Explanation
Problem Analysis We are given the equation of a parabola y 2 = − 4 x . We need to find the value of p , the coordinates of the focus, and the equation of the directrix. The general form of a parabola is y 2 = 4 p x .
Finding the value of p Comparing the given equation y 2 = − 4 x with the general form y 2 = 4 p x , we can find the value of p . We have 4 p = − 4 . Dividing both sides by 4, we get p = − 1 .
Finding the coordinates of the focus Since the equation is of the form y 2 = 4 p x , the vertex of the parabola is at the origin ( 0 , 0 ) . The focus of the parabola y 2 = 4 p x is at ( p , 0 ) . Since p = − 1 , the focus is at ( − 1 , 0 ) .
Finding the equation of the directrix The equation of the directrix for the parabola y 2 = 4 p x is x = − p . Since p = − 1 , the equation of the directrix is x = − ( − 1 ) , which simplifies to x = 1 .
Final Answer Therefore, the value of p is − 1 , the coordinates of the focus are ( − 1 , 0 ) , and the equation of the directrix is x = 1 .
Examples
Understanding parabolas is crucial in various fields like physics and engineering. For instance, satellite dishes and radio telescopes use parabolic reflectors to focus incoming signals to a single point, the focus. The location of the focus and the shape of the parabola are carefully calculated to optimize signal reception. Similarly, the path of a projectile, neglecting air resistance, follows a parabolic trajectory, where understanding the directrix and focus can help predict its range and maximum height.
