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.

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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.

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 \(-\frac{x^2}{4} = 1\).

Multiply both sides of the equation by \(-4\) to isolate \(x^2\): \(x^2 = -4\).

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 \(y=0\).

Therefore, the hyperbola \(\frac{y^2}{2} - \frac{x^2}{4} = 1\) has no x-intercepts, confirming the statement is true.

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주요 개념

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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.

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.