Question

Y=5\ \frac{1}{\left(x-5\right)}+3

A) Parabola
B) Hyperbola
C) Ellipse
D) Circle

Answer 1

B) Hyperbola based on the equation's form and characteristics, it can be confidently identified as a hyperbola.

Explanation:

The given equation
`\(Y = 5 (1)/((x - 5)) + 3\)`is in the form of a hyperbolic function. Hyperbolas are characterized by their equation forms, typically involving the variables x and y in denominators or as a difference between two expressions. In this case, the equation demonstrates a reciprocal relationship between y and (x - 5), creating the characteristic shape of a hyperbola.

The presence of the term
`\(5 (1)/((x - 5))\)` indicates a hyperbolic curve's behavior where the value of y approaches a finite value as x approaches positive or negative infinity. Hence, the equation represents a hyperbola.

Hyperbolas are a type of conic section with distinct features, showcasing two separate curves that diverge away from each other. This equation's structure aligns with the definition and behavior of a hyperbolic function, specifically in the form where x values approach the asymptotes at x = 5. The constant term '3' vertically shifts the hyperbola upwards, but it doesn't change its fundamental nature as a hyperbolic curve. Therefore, based on the equation's form and characteristics, it can be confidently identified as a hyperbola.

The given equation
`\(Y = 5 (1)/((x - 5)) + 3\)`depicts the behavior and structure specific to a hyperbolic function, indicating a hyperbolic curve rather than other conic sections like a parabola, ellipse, or circle.