Question

If one end of the latus rectum of the parabola \(y^2=16x\) is \((4,8)\), then the coordinates of the other end of the latus rectum are

• A. \((4,-16)\)
• B. \((4,10)\)
• C. \((4,-10)\)
• D. \((4,16)\)
• E. \((4,-8)\)

Hint:

For the parabola \(y^2=4ax\), always remember that the endpoints of the latus rectum are \((a,2a)\) and \((a,-2a)\).

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The latus rectum of a parabola is the chord that passes through the focus, is perpendicular to the axis of symmetry, and has its endpoints on the parabola. For a parabola of the form \(y^2 = 4ax\), the axis of symmetry is the x-axis. The endpoints of the latus rectum are therefore symmetric with respect to the x-axis.
Step 2: Key Formula or Approach:
1. Identify the form of the parabola and find the parameter 'a'. The standard form is \(y^2 = 4ax\). 2. The focus of this parabola is at \((a, 0)\). 3. The endpoints of the latus rectum are given by the coordinates \((a, 2a)\) and \((a, -2a)\).
Step 3: Detailed Explanation:
The given equation of the parabola is \(y^2 = 16x\).
Comparing this with the standard form \(y^2 = 4ax\), we get:
\[ 4a = 16 \] \[ a = 4 \] The focus of the parabola is at \((a, 0) = (4, 0)\).
The x-coordinate of both endpoints of the latus rectum is \(x=a=4\).
The y-coordinates of the endpoints are \(y = \pm 2a\).
\[ y = \pm 2(4) = \pm 8 \] So, the two endpoints of the latus rectum are \((4, 8)\) and \((4, -8)\).
We are given that one end is \((4, 8)\). Therefore, the other end must be \((4, -8)\).
Step 4: Final Answer:
The coordinates of the other end of the latus rectum are (4, -8). This corresponds to option (E).