Question

If \( (2,3) \) is the focus and \( x - y + 3 = 0 \) is the directrix of a parabola, then the equation of the tangent drawn at the vertex of the parabola is:

• A. \( x - y - 2 = 0 \)
• B. \( x - y + 2 = 0 \)
• C. \( x - y + 5 = 0 \)
• D. \( x - y - 5 = 0 \)

Hint:

The vertex of a parabola is the midpoint between the focus and directrix.

Answer 1

The Correct Option is B

Step 1: Compute the vertex of the parabola.
The vertex is the midpoint of the focus and the directrix: \[ V = \left( \frac{2 + x}{2}, \frac{3 + y}{2} \right) \] Solving for \( x, y \) using the directrix equation, we find: \[ V = (2, 1) \] Step 2: Finding the equation of the tangent at the vertex.
The tangent at the vertex is perpendicular to the axis of symmetry. The axis of symmetry is the perpendicular bisector of the focus and directrix, whose slope is: \[ 1 \] Thus, the equation of the tangent is: \[ x - y + 2 = 0 \]