Question

For the parabola represented parametrically by \( x = t^2 + t + 1 \) and \( y = t^2 - t + 1 \), the length of the latus rectum is:

• A. \( 3 \)
• B. \( 2 \)
• C. \( \frac{1}{2} \)
• D. \( 8 \)

Hint:

Parametric Form and Latus Rectum}
Find relationship between \( x \) and \( y \)
Express in terms of standard parabolic form
Recall: Latus rectum = \( 4a \) for \( y^2 = 4ax \)

Answer 1

The Correct Option is B

We are given parametric equations: \[ x = t^2 + t + 1,\quad y = t^2 - t + 1 \Rightarrow x - y = 2t \Rightarrow t = \frac{x - y}{2} \] Now substitute into \( x \): \[ x = \left( \frac{x - y}{2} \right)^2 + \frac{x - y}{2} + 1 \Rightarrow \text{This represents a parabola} \] This parabola can be reduced to standard form via rotation/translation, but since it's derived from general parabola form, it has standard latus rectum length = 4a. From form: \[ (x - y)^2 = 4a(x + y) \Rightarrow \text{Latus rectum length} = 4a = 2 \]