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Which of the following is the equation for the graph shown?

graph of vertical hyperbola on a coordinate plane going through about 2 and one fourth comma 0 and negative 2 and one fourth comma 0 with points at negative 3 comma 0 and 3 comma 0, lines at y equals five thirds and y equals negative five thirds

y squared over 4 minus x squared over 5 equals 1
x squared over 4 minus y squared over 5 equals 1
x squared over 5 minus y squared over 4 equals 1
y squared over 5 minus x squared over 4 equals 1

Respuesta :

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

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