Applications

Conic Identification

Classify an axis-aligned conic from its squared-term coefficients.

conic-identification With no xy term, compare the x^2 and y^2 coefficients. Same equal signs make a circle. Same unequal signs make an ellipse. One squared term makes a parabola. Opposite signs make a hyperbola.

Example

Use coefficient signs to classify an axis-aligned conic.

highlighted = computed this step

Step 1 — Read the equation

Read the squared-term coefficients.

9x2+25y2=225\hl{9} x^{2} + \hl{25} y^{2} = \hl{225}

Step 2 — Compare coefficients

Compare the x and y coefficients.

A=9B=25A= \hl{9} \quad B= \hl{25}

Step 3 — Use the sign rule

Use the signs to choose the conic family.

same sign and unequal coefficients\text{same sign and unequal coefficients}

Step 4 — Classify

State the conic classification.

conic type: ellipse\text{conic type: ellipse}

Step 5 — Plot the conic

Plot the identified conic.

Coordinate plot\text{Coordinate plot}
Conic identification plotThe equation is an ellipse centered at the origin.

Step 6 — Verdict

State the classification verdict.

classification: ellipse\text{classification: ellipse}