Consider the equation $y=(x+2)^2-3$. What is the length of the focal width of the parabola?
A. 1 unit
B. 4 units
C. 12 units
D. 16 units
Answer (2)
Rewrite the given equation y = ( x + 2 ) 2 − 3 in the standard form: ( x + 2 ) 2 = y + 3 .
Compare the equation with the standard form ( x − h ) 2 = 4 p ( y − k ) to find 4 p = 1 .
The length of the focal width is ∣4 p ∣ = 1 .
The length of the focal width of the parabola is 1 unit.
Explanation
Understanding the Problem We are given the equation of a parabola y = ( x + 2 ) 2 − 3 and asked to find the length of its focal width (also known as the latus rectum). The focal width is the length of the line segment through the focus of the parabola, perpendicular to the axis of symmetry, with endpoints on the parabola.
Standard Form of a Parabola To find the focal width, we need to rewrite the equation in the standard form of a parabola. The standard form for a parabola that opens upwards or downwards is ( x − h ) 2 = 4 p ( y − k ) , where ( h , k ) is the vertex of the parabola and p is the distance from the vertex to the focus. The length of the focal width (latus rectum) is given by ∣4 p ∣ .
Rewriting the Equation Let's rewrite the given equation y = ( x + 2 ) 2 − 3 in the standard form. Adding 3 to both sides, we get y + 3 = ( x + 2 ) 2 , which can be written as ( x + 2 ) 2 = y + 3 .
Identifying the Parameters Comparing the equation ( x + 2 ) 2 = y + 3 with the standard form ( x − h ) 2 = 4 p ( y − k ) , we can identify the vertex as ( h , k ) = ( − 2 , − 3 ) . Also, we can see that 4 p = 1 .
Finding the Focal Width The length of the focal width is ∣4 p ∣ = ∣1∣ = 1 . Therefore, the length of the focal width of the parabola is 1 unit.
Examples
Understanding the focal width of a parabola is useful in various applications, such as designing satellite dishes or analyzing the trajectory of projectiles. For example, if you are designing a satellite dish, the shape of the dish is parabolic, and the receiver is placed at the focus of the parabola. The focal width helps determine the size and shape of the receiver needed to capture the signal effectively. Similarly, in physics, the path of a projectile under constant gravitational force is a parabola, and understanding its focal width can help predict its range and trajectory.
The length of the focal width of the parabola given by the equation y = ( x + 2 ) 2 − 3 is 1 unit. This is found by rewriting the equation in standard form and recognizing that 4 p = 1 . Thus, the answer is A. 1 unit.
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