The equation for the graph is obtained by making use of the relationship
between the directrix and the eccentricity.
Correct response:
- [tex]The \ equation \ of \ the \ graph \ is ; \ \underline{\dfrac{y^2}{5} - \dfrac{x^2}{4} = 1}[/tex]
Method by which the above equation is found
The general form of the equation of a vertical hyperbola is given as follows;
[tex]\mathbf{\dfrac{(y - k)^2}{a^2} - \dfrac{(x - h)^2}{b^2}} = 1[/tex]
From the given options, the center of the hyperbola, (h, k) = (0, 0)
The points on the hyperbola are;
[tex]\left(2\frac{1}{4} , \ 0 \right)[/tex], [tex]\left(-2\frac{1}{4} , \, 0 \right )[/tex]
(-3, 0) and (3, 0)
The given directrices are;
[tex]y = \frac{5}{3}[/tex] and [tex]y = -\frac{5}{3}[/tex]
- [tex]Directrix, \ y = \mathbf{\pm \dfrac{a}{e}}[/tex]
- [tex]Eccentricity, \ e = \mathbf{ \dfrac{\sqrt{a^2 + b^2} }{a}}[/tex]
Therefore;
- [tex]Directrix, \, y = \mathbf{\dfrac{a^2}{\sqrt{a^2 + b^2} }}[/tex]
We have;
a² = 5
√(a² + b²) = 3
Therefore;
5 + b² = 9
b² = 4
- Which gives the equation of the parabola as [tex]\underline{\dfrac{y^2}{5} - \dfrac{x^2}{4} = 1}[/tex], which is the option;
- y squared over 5 minus x squared over 4 equals 1
Learn more about a hyperbola here:
https://brainly.com/question/2364331
