Question

find the focus and the equation of the directrix of the parabola 2 X square + 3 x minus 2 Y - 1 is equal to zero
​

Answer 1

Focus is $(- \frac{3}{4}, - \frac{17}{16})$ and the directrix is $y = - \frac{21}{16}$ ,

Step-by-step explanation:

The equation of the parabola is 2x² + 3x - 2y - 1 = 0

Arranging the formula in vertex form we get,

2x² + 3x - 1 = 2y

⇒ $2[x^{2} + 2(x)(\frac{3}{4}) + (\frac{3}{4} )^{2}] - 1 - \frac{9}{8} = 2y$

⇒ $2(x + \frac{3}{4})^{2} = 2y + \frac{17}{8}$

⇒ $(x + \frac{3}{4})^{2} = y + \frac{17}{16}$

This equation is comparable with the equation (x - α)² = 4a(y - β).

Therefore, the parabola has an axis parallel to the positive y-axis the vertex at $(- \frac{3}{4}, - \frac{17}{16})$ and a = $\frac{1}{4}$.

Now, the focus of the parabola is $\frac{1}{4}$ distance from vertex along parallel to the y-axis.

So, the focus will be $(- \frac{3}{4}, - \frac{17}{16} + \frac{1}{4} )$ = $(- \frac{3}{4}, - \frac{13}{16})$.

And  $y = - \frac{17}{16} - \frac{1}{4} = - \frac{21}{16}$ is the equation of the directrix. (Answer)