Question

For the parabola represented in the parametric form \[ x=t^2+t+1 \] \[ y=t^2-t+1, \] the length of latus rectum is:

• A. \(2\)
• B. \(3\)
• C. \(\frac12\)
• D. \(8\)

Hint:

Convert the parametric equations into a Cartesian equation and compare with the standard parabola \[ X^2=4aY. \] The latus rectum length is always \(4a\).

Answer 1

The Correct Option is A

Step 1: Eliminate the parameter.
\(x-y=2t\) and \(x+y=2t^2+2\). So \(t=\frac{x-y}{2}\) and \(x+y-2=2\left(\frac{x-y}{2}\right)^2=\frac{(x-y)^2}{2}\).

Step 2: Identify latus rectum.
Let \(u=x+y-2\), \(v=x-y\): equation \(v^2=2u\) is a parabola in the form \(V^2=4aU\) with \(4a=2\Rightarrow a=\frac{1}{2}\). Latus rectum \(=4a=\boxed{2}\).