Question

If one end of a focal chord of the parabola, $y^2 = 16x$ is at $(1, 4),$ then the length of this focal chord is

• A. 25
• B. 24
• C. 20
• D. 22

Answer 1

The Correct Option is A

To find the length of the focal chord of the parabola \(y^2 = 16x\), let's first understand some properties of a parabola and how to calculate the length of a focal chord.

The general equation of a parabola in the form \(y^2 = 4ax\) represents a parabola that opens to the right. Here, it is given as \(y^2 = 16x\), which means \(4a = 16\). Thus, the value of \(a\) is:

\(a = \frac{16}{4} = 4\)

This represents a parabola with vertex at the origin \((0, 0)\) and focus at \((a, 0) = (4, 0)\).

Given that the point \((1, 4)\) is one end of the focal chord, we need to find the other end and calculate the length of the chord.

For a parabola, a focal chord is a line segment through the focus that terminates on the parabola. If one end of the focal chord is \((x_1, y_1) = (1, 4)\), we use the parametric equations of the parabola to determine the coordinates of the other end.

The parametric form of a point on the parabola \(y^2 = 4ax\) is \((at^2, 2at)\). We equate this with \((1, 4)\):

• \(at_1^2 = 1\) implies \(t_1^2 \cdot 4 = 1\) leading to \(t_1^2 = \frac{1}{4}\) hence \(t_1 = \pm \frac{1}{2}\).
• \(2at_1 = 4\) implies \(2 \cdot 4 \cdot t_1 = 4\) hence \(t_1 = \frac{1}{2}\).

Next, calculate the corresponding coordinates for the other end of the focal chord using \(t_2 = - \frac{1}{t_1}\):

• \(t_2 = -2\), \text{so the coordinates are} : \((at_2^2, 2at_2) = (4 \cdot 4, -16) = (16, -16)\).

Now, we have both ends of the chord \((1, 4)\) and \((16, -16)\). The length of a focal chord of the parabola is given by the distance between these two points:

Let's use the distance formula:

\(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(16 - 1)^2 + (-16 - 4)^2}\)

\(= \sqrt{15^2 + (-20)^2} = \sqrt{225 + 400} = \sqrt{625} = 25\)

Therefore, the length of this focal chord is \(\textbf{25}\).