which equations graph is a hyperbola? \n○ $y^2 = 25x$ \n○ $x^2 - 25y^2 = 225$ \n○ $x + 25y = 225$ \n○ $x^2 +…

which equations graph is a hyperbola? \n○ $y^2 = 25x$ \n○ $x^2 - 25y^2 = 225$ \n○ $x + 25y = 225$ \n○ $x^2 + 25y^2 = 225$

which equations graph is a hyperbola? \n○ $y^2 = 25x$ \n○ $x^2 - 25y^2 = 225$ \n○ $x + 25y = 225$ \n○ $x^2 + 25y^2 = 225$

Answer

Explanation:

Step1: Recall Conic Section Forms

The standard form of a hyperbola is (\frac{x^2}{a^2}-\frac{y^2}{b^2}=1) or (\frac{y^2}{a^2}-\frac{x^2}{b^2}=1) (difference of squares). For an ellipse, it's (\frac{x^2}{a^2}+\frac{y^2}{b^2}=1) (sum of squares). A parabola has one squared term (e.g., (y^2 = 4ax) or (x^2=4ay)), and a line is linear (no squared terms).

Step2: Analyze Each Option

  • Option 1: (y^2 = 25x) – This is a parabola (only (y^2) term, linear in (x)).
  • Option 2: (x^2 - 25y^2 = 225) – Rewrite as (\frac{x^2}{225}-\frac{y^2}{9}=1) (divide by 225), which matches the hyperbola form (difference of (x^2) and (y^2) terms).
  • Option 3: (x + 25y = 225) – Linear equation (no squared terms), represents a line.
  • Option 4: (x^2 + 25y^2 = 225) – Rewrite as (\frac{x^2}{225}+\frac{y^2}{9}=1), which is an ellipse (sum of (x^2) and (y^2) terms).

Answer:

The equation whose graph is a hyperbola is (x^2 - 25y^2 = 225) (the second option in the list).