Question

The equation of the directrix of the parabola \(x^{2}-4x-5y-1=0\) is...

• A. \(4x+9=0\)
• B. \(4y+9=0\)
• C. \(4y-9=0\)
• D. \(4x-9=0\)

Hint:

Complete the square to reach the form (x-h)^2 = 4a(y-k), and then the directrix is y = k - a.

Answer 1

The Correct Option is B

Step 1: Shift the Origin
Let $X=x-2$ and $Y=y+1$; the equation becomes $X^{2}=5Y$.

Step 2: Directrix in New Axes
For $X^{2}=4aY$ with $a=\dfrac{5}{4}$, the directrix is $Y=-\dfrac{5}{4}$.

Step 3: Back to the Original
$y+1=-\dfrac{5}{4}\Rightarrow 4y+9=0$, option (B).

Final Answer:
The directrix is the line $4y+9=0$, option (B). \[ \boxed{\text{(B) }4y+9=0} \]