An equation in standard form for a hyperbola with center (0, 0), vertex (-5, 0), and focus (-6, 0) is given by y²/25 - x²/9 = 1.
<h3>What is an equation?</h3>
An equation can be defined as a mathematical expression which is used to show and indicate that two (2) or more numerical quantities are equal.
<h3>How to determine the equation of a hyperbola?</h3>
Mathematically, the equation of a hyperbola in standard form is given by:
![\frac{(y\;-\;k)^2}{a^2} - \frac{(x\;-\;h)^2}{b^2} = 1](https://tex.z-dn.net/?f=%5Cfrac%7B%28y%5C%3B-%5C%3Bk%29%5E2%7D%7Ba%5E2%7D%20-%20%5Cfrac%7B%28x%5C%3B-%5C%3Bh%29%5E2%7D%7Bb%5E2%7D%20%3D%201)
<u>Given the following data:</u>
Center (h, k) = (0, 0)
Vertex (h+a, k) = (-5, 0)
Foci, F = (h+c, k) = (-6, 0) and F' = (6, 0)
Also, we can logically deduce that the value of a and c are -5 and -6 respectively.
For the value of b, we would apply Pythagorean's theorem:
c² = a² + b²
b² = c² - a²
b² = (-6)² - (-5)²
b² = 36 - 25
b² = 9.
b = √9
b = 3.
Substituting the given parameters into the equation of a hyperbola in standard form, we have;
![\frac{(y\;-\;k)^2}{a^2} - \frac{(x\;-\;h)^2}{b^2} = 1\\\\\frac{(y\;-\;0)^2}{-5^2} - \frac{(x\;-\;0)^2}{3^2} = 1\\\\\frac{y^2}{-5^2} - \frac{x^2}{3^2} = 1\\\\\frac{y^2}{25} - \frac{x^2}{9} = 1](https://tex.z-dn.net/?f=%5Cfrac%7B%28y%5C%3B-%5C%3Bk%29%5E2%7D%7Ba%5E2%7D%20-%20%5Cfrac%7B%28x%5C%3B-%5C%3Bh%29%5E2%7D%7Bb%5E2%7D%20%3D%201%5C%5C%5C%5C%5Cfrac%7B%28y%5C%3B-%5C%3B0%29%5E2%7D%7B-5%5E2%7D%20-%20%5Cfrac%7B%28x%5C%3B-%5C%3B0%29%5E2%7D%7B3%5E2%7D%20%3D%201%5C%5C%5C%5C%5Cfrac%7By%5E2%7D%7B-5%5E2%7D%20-%20%5Cfrac%7Bx%5E2%7D%7B3%5E2%7D%20%3D%201%5C%5C%5C%5C%5Cfrac%7By%5E2%7D%7B25%7D%20-%20%5Cfrac%7Bx%5E2%7D%7B9%7D%20%3D%201)
y²/25 - x²/9 = 1.
In conclusion, we can logically deduce that an equation in standard form for a hyperbola with center (0, 0), vertex (-5, 0), and focus (-6, 0) is given by y²/25 - x²/9 = 1.
Read more on hyperbola here: brainly.com/question/3405939
#SPJ1