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MATH / The equation of the hyperbola shown below can be written as

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

The equation of the hyperbola shown below can be written as (y−k)2a2−(x−h)2b2=1.\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1.Find h+k+a+b.h + k + a + b. [asy] unitsize(0.3 cm); real upperhyper (real x) { return (2*sqrt((x - 6)^2/16 + 1) - 5); } real lowerhyper (real x) { return (-2*sqrt((x - 6)^2/16 + 1) - 5); } int i, n = 10; for (i = -n; i <= n; ++i) { draw((i,-n)--(i,n),gray(0.7)); draw((-n,i)--(n,i),gray(0.7)); } draw((0,-n)--(0,n)); draw((-n,0)--(n,0)); draw(graph(upperhyper,-10,10),red); draw(graph(lowerhyper,-3,10),red); draw(extension((-10,-10),(10,-10),(6,-5),(6,-5) + (4,2))--extension((10,10),(10,-10),(6,-5),(6,-5) + (4,2)),dashed); draw(extension((-10,-10),(-10,10),(6,-5),(6,-5) + (4,-2))--extension((10,10),(10,-10),(6,-5),(6,-5) + (4,-2)),dashed); dot((6,-5)); [/asy]
Plain-text mathematical notation (without MathML)
The equation of the hyperbola shown below can be written as
((y−k)²)/(a²)−((x−h)²)/(b²)=1.Find h+k+a+b.

[asy]
unitsize(0.3 cm);

real upperhyper (real x) {
  return (2*sqrt((x - 6)^2/16 + 1) - 5);
}

real lowerhyper (real x) {
  return (-2*sqrt((x - 6)^2/16 + 1) - 5);
}

int i, n = 10;

for (i = -n; i <= n; ++i) {
  draw((i,-n)--(i,n),gray(0.7));
  draw((-n,i)--(n,i),gray(0.7));
}

draw((0,-n)--(0,n));
draw((-n,0)--(n,0));

draw(graph(upperhyper,-10,10),red);
draw(graph(lowerhyper,-3,10),red);
draw(extension((-10,-10),(10,-10),(6,-5),(6,-5) + (4,2))--extension((10,10),(10,-10),(6,-5),(6,-5) + (4,2)),dashed);
draw(extension((-10,-10),(-10,10),(6,-5),(6,-5) + (4,-2))--extension((10,10),(10,-10),(6,-5),(6,-5) + (4,-2)),dashed);

dot((6,-5));
[/asy]
Original LaTeX notation
The equation of the hyperbola shown below can be written as
\[\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1.\]Find $h + k + a + b.$

[asy]
unitsize(0.3 cm);

real upperhyper (real x) {
  return (2*sqrt((x - 6)^2/16 + 1) - 5);
}

real lowerhyper (real x) {
  return (-2*sqrt((x - 6)^2/16 + 1) - 5);
}

int i, n = 10;

for (i = -n; i <= n; ++i) {
  draw((i,-n)--(i,n),gray(0.7));
  draw((-n,i)--(n,i),gray(0.7));
}

draw((0,-n)--(0,n));
draw((-n,0)--(n,0));

draw(graph(upperhyper,-10,10),red);
draw(graph(lowerhyper,-3,10),red);
draw(extension((-10,-10),(10,-10),(6,-5),(6,-5) + (4,2))--extension((10,10),(10,-10),(6,-5),(6,-5) + (4,2)),dashed);
draw(extension((-10,-10),(-10,10),(6,-5),(6,-5) + (4,-2))--extension((10,10),(10,-10),(6,-5),(6,-5) + (4,-2)),dashed);

dot((6,-5));
[/asy]

level

Level 4

type

Intermediate Algebra

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Source and history

task_id
deedd79d-70a8-54e0-af4f-dcdbc0b3f455
task_key
intermediate~5falgebra--test--deedd79d-70a8-54e0-af4f-dcdbc0b3f455
task_revision_id
1
config / split
intermediate_algebra / test

Official source

initial import