THE NUMORIX GUIDE
How to use the Hyperbola Equation Calculator
Last reviewed September 14, 2026
What this calculator does
For the horizontal standard form (x-h)^2/a^2 - (y-k)^2/b^2 = 1, calculate c=sqrt(a^2+b^2), foci at (h +/- c,k), eccentricity e=c/a, and asymptotes y-k = +/- (b/a)(x-h).
Formula and method
For the horizontal standard form (x-h)^2/a^2 - (y-k)^2/b^2 = 1, calculate c=sqrt(a^2+b^2), foci at (h +/- c,k), eccentricity e=c/a, and asymptotes y-k = +/- (b/a)(x-h).
Variables and inputs
Center defaults to (0,0), semi-transverse axis a=3, and semi-conjugate axis b=4. The implementation models only a horizontal hyperbola opening left and right.
Worked example
For h=k=0,a=3,b=4: equation x^2/9-y^2/16=1; c=sqrt(9+16)=5; foci=(-5,0),(5,0); e=5/3=1.6667; asymptotes y=+/-4/3*x.
How to interpret the result
The foci describe the distance-difference property of a hyperbola; asymptotes show the lines the branches approach without reaching.
Common mistakes to avoid
Use a^2+b^2 for c in a hyperbola, not a^2-b^2 from an ellipse. Keep the minus sign between the squared terms and the center shifts consistent.
Assumptions and limitations
Only the horizontal orientation is implemented, and positive-axis validation is not explicit. The displayed asymptotes are rounded to four decimal places.
Practical use and checks
The Hyperbola Equation Calculator builds a horizontal hyperbola from center, semi-transverse axis a, and semi-conjugate axis b. Enter center (0, 0), a = 3, and b = 4 as a check; the equation should be x^2/9 - y^2/16 = 1, focal distance c = 5, foci (-5, 0) and (5, 0), eccentricity about 1.6667, and asymptotes y = plus or minus 4x/3. The relation c^2 = a^2 + b^2 is a useful check on the focus calculation. Use the asymptotes to understand the branches and to sketch behavior far from the center; they are not part of the curve itself. The transverse axis and conjugate axis have different roles, so swapping a and b changes the equation and the asymptote slopes. The calculator assumes a horizontal, nonrotated, ideal hyperbola with positive axis inputs. It does not represent a vertical or rotated hyperbola, a shifted coordinate transformation beyond the center fields, uncertainty, or a finite physical boundary. Large or nearly zero axes can make the displayed slopes sensitive to rounding. For an engineering or data-fitting use, verify the orientation and coordinate convention before treating a point as on a branch, and remember that an asymptote is approached rather than crossed as a geometric boundary in the basic model.