Hyperbolas

Textbook reading

The problem that opens the lesson

A hyperbola has center $(0,0),$ vertices $(\pm 3,0),$ and foci $(\pm 5,0)$ . Find its equation and asymptotes.

Solution

Begin by identifying the mathematical object and the information that fixes it. Determine center and orientation, locate vertices, compute c, draw the fundamental rectangle, and sketch branches approaching the asymptotes. The relevant conditions are not optional bookkeeping: The two branches are separate components of one relation. The asymptotes guide end behavior but are not part of the hyperbola. Following that structure gives $a=3,c=5,b=4$ ; $\frac{x^2}{9}-\frac{y^2}{16}=1$ ; $y=\frac{\pm 4x}{3}$ .

Why this works

Asymptotes pass through the center and follow the diagonals of the fundamental rectangle with semiaxes a and $b$ . The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Textbook reading

What this lesson is really about

A hyperbola is the locus of points whose absolute difference of distances to two foci is constant $2a$ .

Its standard forms contain one positive squared term and one negative squared term. The positive term identifies the transverse-axis direction. The focal relationship is $c^2=a^2+b^2$ .

The point is not merely to reproduce a formula. A learner should be able to identify the quantities or geometric objects involved, explain why the relationship has its stated form, and recognize when the same idea appears in a graph, table, diagram, or model.

Textbook reading

Why the relationship works

Asymptotes pass through the center and follow the diagonals of the fundamental rectangle with semiaxes a and $b$ .

Textbook reading

A reliable way to work

Determine center and orientation, locate vertices, compute c, draw the fundamental rectangle, and sketch branches approaching the asymptotes.

The two branches are separate components of one relation. The asymptotes guide end behavior but are not part of the hyperbola.

After the symbolic work is complete, check the result. Depending on the lesson, this may mean substituting into an original equation, comparing coordinates, examining a graph, checking units, testing an interval, or confirming that every branch of a periodic solution has been included.

Textbook reading

What commonly goes wrong

A common error is using the ellipse relation $c^2=a^2-b^2$ or orienting the branches by the larger denominator rather than the positive term.

The repair is to return to the definition and identify the first step where the invalid solution stops describing the original mathematical object. Later algebra cannot rescue a first step that changed the domain, orientation, branch, or meaning of the problem.

Textbook reading

Worked examples

Worked example 1

A hyperbola has center $(0,0),$ vertices $(\pm 3,0),$ and foci $(\pm 5,0)$ . Find its equation and asymptotes.

Solution

Begin by identifying the mathematical object and the information that fixes it. Determine center and orientation, locate vertices, compute c, draw the fundamental rectangle, and sketch branches approaching the asymptotes. The relevant conditions are not optional bookkeeping: The two branches are separate components of one relation. The asymptotes guide end behavior but are not part of the hyperbola. Following that structure gives $a=3,c=5,b=4$ ; $\frac{x^2}{9}-\frac{y^2}{16}=1$ ; $y=\frac{\pm 4x}{3}$ .

Why this works

Asymptotes pass through the center and follow the diagonals of the fundamental rectangle with semiaxes a and $b$ . The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Transfer example

Problem

Derive $c^2=a^2+b^2$ .

Worked development

Determine center and orientation, locate vertices, compute c, draw the fundamental rectangle, and sketch branches approaching the asymptotes. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. Its standard forms contain one positive squared term and one negative squared term. The positive term identifies the transverse-axis direction. The focal relationship is $c^2=a^2+b^2$ . Then apply the conditions explicitly: The two branches are separate components of one relation. The asymptotes guide end behavior but are not part of the hyperbola. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

Hyperbolas appear in navigation differences, cooling-tower sections, optics, and inverse-product graphs after transformation.

Reasoning example

Problem

Read transverse and conjugate axes.

Worked development

Determine center and orientation, locate vertices, compute c, draw the fundamental rectangle, and sketch branches approaching the asymptotes. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. Its standard forms contain one positive squared term and one negative squared term. The positive term identifies the transverse-axis direction. The focal relationship is $c^2=a^2+b^2$ . Then apply the conditions explicitly: The two branches are separate components of one relation. The asymptotes guide end behavior but are not part of the hyperbola. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

Hyperbolas appear in navigation differences, cooling-tower sections, optics, and inverse-product graphs after transformation.

Worked example 4: quick check

Find vertices and asymptotes of $\frac{y^2}{25}-\frac{x^2}{9}=1$

Solution

Begin by identifying the mathematical object and the information that fixes it. Determine center and orientation, locate vertices, compute c, draw the fundamental rectangle, and sketch branches approaching the asymptotes. The relevant conditions are not optional bookkeeping: The two branches are separate components of one relation. The asymptotes guide end behavior but are not part of the hyperbola. Following that structure gives Vertices $(0,\pm 5)$ ; asymptotes $y=\pm (\frac{5}{3})x$ .

Why this works

Asymptotes pass through the center and follow the diagonals of the fundamental rectangle with semiaxes a and $b$ . The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Textbook reading

Application and interpretation

Hyperbolas appear in navigation differences, cooling-tower sections, optics, and inverse-product graphs after transformation.

A contextual answer must include units, a meaningful domain, and the assumptions that make the model plausible. An exact mathematical relationship should not be diluted into a decimal unless a measurement or comparison requires it.

Practice