Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
e. The hyperbola y²/2 - x²/4 = 1 has no x-intercept.

Verified step by step guidance

1. Recall that the x-intercepts of a curve occur where the graph crosses the x-axis, which means the y-coordinate is zero. So, to find the x-intercepts, set \(y = 0\) in the equation of the hyperbola.
2. Substitute \(y = 0\) into the equation \(\frac{y^2}{2} - \frac{x^2}{4} = 1\), which simplifies to \(\frac{0^2}{2} - \frac{x^2}{4} = 1\), or \(-\frac{x^2}{4} = 1\).
3. Multiply both sides of the equation by \(-4\) to isolate \(x^2\): \(x^2 = -4\).
4. Since \(x^2 = -4\) has no real solutions (because the square of a real number cannot be negative), there are no real values of \(x\) that satisfy the equation when \(y=0\).
5. Therefore, the hyperbola \(\frac{y^2}{2} - \frac{x^2}{4} = 1\) has no x-intercepts, confirming the statement is true.

Key Concepts

Definition of x-intercept

An x-intercept is a point where a graph crosses the x-axis, meaning the y-coordinate is zero. To find x-intercepts, set y = 0 in the equation and solve for x. If no real solutions exist, the graph has no x-intercepts.

Equation of a hyperbola

A hyperbola is a conic section defined by an equation of the form (y²/a²) - (x²/b²) = 1 or (x²/a²) - (y²/b²) = 1. It represents two separate curves and its shape depends on the signs and values of a² and b².

Solving for intercepts in conic sections

To determine intercepts in conic sections, substitute the corresponding coordinate (x=0 for y-intercept, y=0 for x-intercept) into the equation. Analyze the resulting equation for real solutions to confirm the existence of intercepts.