Algebra Examples

Find the Hyperbola: Center (5,6), Focus (-5,6), Vertex (4,6) (5,6) , (4,6) , (-5,6)
(5,6)(5,6) , (4,6)(4,6) , (-5,6)(5,6)
Step 1
There are two general equations for a hyperbola.
Horizontal hyperbola equation (x-h)2a2-(y-k)2b2=1(xh)2a2(yk)2b2=1
Vertical hyperbola equation (y-k)2a2-(x-h)2b2=1(yk)2a2(xh)2b2=1
Step 2
aa is the distance between the vertex (4,6)(4,6) and the center point (5,6)(5,6).
Tap for more steps...
Step 2.1
Use the distance formula to determine the distance between the two points.
Distance=(x2-x1)2+(y2-y1)2Distance=(x2x1)2+(y2y1)2
Step 2.2
Substitute the actual values of the points into the distance formula.
a=(4-5)2+(6-6)2a=(45)2+(66)2
Step 2.3
Simplify.
Tap for more steps...
Step 2.3.1
Subtract 55 from 44.
a=(-1)2+(6-6)2a=(1)2+(66)2
Step 2.3.2
Raise -11 to the power of 22.
a=1+(6-6)2a=1+(66)2
Step 2.3.3
Subtract 66 from 66.
a=1+02a=1+02
Step 2.3.4
Raising 00 to any positive power yields 00.
a=1+0a=1+0
Step 2.3.5
Add 11 and 00.
a=1a=1
Step 2.3.6
Any root of 11 is 11.
a=1a=1
a=1a=1
a=1a=1
Step 3
cc is the distance between the focus (-5,6)(5,6) and the center (5,6)(5,6).
Tap for more steps...
Step 3.1
Use the distance formula to determine the distance between the two points.
Distance=(x2-x1)2+(y2-y1)2Distance=(x2x1)2+(y2y1)2
Step 3.2
Substitute the actual values of the points into the distance formula.
c=((-5)-5)2+(6-6)2c=((5)5)2+(66)2
Step 3.3
Simplify.
Tap for more steps...
Step 3.3.1
Subtract 55 from -55.
c=(-10)2+(6-6)2c=(10)2+(66)2
Step 3.3.2
Raise -1010 to the power of 22.
c=100+(6-6)2c=100+(66)2
Step 3.3.3
Subtract 66 from 66.
c=100+02c=100+02
Step 3.3.4
Raising 00 to any positive power yields 00.
c=100+0c=100+0
Step 3.3.5
Add 100100 and 00.
c=100c=100
Step 3.3.6
Rewrite 100100 as 102102.
c=102c=102
Step 3.3.7
Pull terms out from under the radical, assuming positive real numbers.
c=10c=10
c=10c=10
c=10c=10
Step 4
Using the equation c2=a2+b2c2=a2+b2. Substitute 11 for aa and 1010 for cc.
Tap for more steps...
Step 4.1
Rewrite the equation as (1)2+b2=102(1)2+b2=102.
(1)2+b2=102(1)2+b2=102
Step 4.2
One to any power is one.
1+b2=1021+b2=102
Step 4.3
Raise 1010 to the power of 22.
1+b2=1001+b2=100
Step 4.4
Move all terms not containing bb to the right side of the equation.
Tap for more steps...
Step 4.4.1
Subtract 11 from both sides of the equation.
b2=100-1b2=1001
Step 4.4.2
Subtract 11 from 100100.
b2=99b2=99
b2=99b2=99
Step 4.5
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
b=±99b=±99
Step 4.6
Simplify ±99±99.
Tap for more steps...
Step 4.6.1
Rewrite 9999 as 32113211.
Tap for more steps...
Step 4.6.1.1
Factor 99 out of 9999.
b=±9(11)b=±9(11)
Step 4.6.1.2
Rewrite 99 as 3232.
b=±3211b=±3211
b=±3211b=±3211
Step 4.6.2
Pull terms out from under the radical.
b=±311b=±311
b=±311b=±311
Step 4.7
The complete solution is the result of both the positive and negative portions of the solution.
Tap for more steps...
Step 4.7.1
First, use the positive value of the ±± to find the first solution.
b=311b=311
Step 4.7.2
Next, use the negative value of the ±± to find the second solution.
b=-311b=311
Step 4.7.3
The complete solution is the result of both the positive and negative portions of the solution.
b=311,-311b=311,311
b=311,-311b=311,311
b=311,-311b=311,311
Step 5
bb is a distance, which means it should be a positive number.
b=311b=311
Step 6
The slope of the line between the focus (-5,6)(5,6) and the center (5,6)(5,6) determines whether the hyperbola is vertical or horizontal. If the slope is 00, the graph is horizontal. If the slope is undefined, the graph is vertical.
Tap for more steps...
Step 6.1
Slope is equal to the change in yy over the change in xx, or rise over run.
m=change in ychange in xm=change in ychange in x
Step 6.2
The change in xx is equal to the difference in x-coordinates (also called run), and the change in yy is equal to the difference in y-coordinates (also called rise).
m=y2-y1x2-x1m=y2y1x2x1
Step 6.3
Substitute in the values of xx and yy into the equation to find the slope.
m=6-(6)5-(-5)m=6(6)5(5)
Step 6.4
Simplify.
Tap for more steps...
Step 6.4.1
Simplify the numerator.
Tap for more steps...
Step 6.4.1.1
Multiply -11 by 66.
m=6-65-(-5)m=665(5)
Step 6.4.1.2
Subtract 66 from 66.
m=05-(-5)m=05(5)
m=05-(-5)m=05(5)
Step 6.4.2
Simplify the denominator.
Tap for more steps...
Step 6.4.2.1
Multiply -11 by -55.
m=05+5m=05+5
Step 6.4.2.2
Add 55 and 55.
m=010m=010
m=010m=010
Step 6.4.3
Divide 00 by 1010.
m=0m=0
m=0m=0
Step 6.5
The general equation for a horizontal hyperbola is (x-h)2a2-(y-k)2b2=1(xh)2a2(yk)2b2=1.
(x-h)2a2-(y-k)2b2=1(xh)2a2(yk)2b2=1
(x-h)2a2-(y-k)2b2=1(xh)2a2(yk)2b2=1
Step 7
Substitute the values h=5h=5, k=6k=6, a=1a=1, and b=311b=311 into (x-h)2a2-(y-k)2b2=1(xh)2a2(yk)2b2=1 to get the hyperbola equation (x-(5))2(1)2-(y-(6))2(311)2=1(x(5))2(1)2(y(6))2(311)2=1.
(x-(5))2(1)2-(y-(6))2(311)2=1(x(5))2(1)2(y(6))2(311)2=1
Step 8
Simplify to find the final equation of the hyperbola.
Tap for more steps...
Step 8.1
Multiply -11 by 55.
(x-5)212-(y-(6))2(311)2=1(x5)212(y(6))2(311)2=1
Step 8.2
One to any power is one.
(x-5)21-(y-(6))2(311)2=1(x5)21(y(6))2(311)2=1
Step 8.3
Divide (x-5)2(x5)2 by 11.
(x-5)2-(y-(6))2(311)2=1(x5)2(y(6))2(311)2=1
Step 8.4
Multiply -11 by 66.
(x-5)2-(y-6)2(311)2=1(x5)2(y6)2(311)2=1
Step 8.5
Simplify the denominator.
Tap for more steps...
Step 8.5.1
Apply the product rule to 311311.
(x-5)2-(y-6)232112=1(x5)2(y6)232112=1
Step 8.5.2
Raise 33 to the power of 22.
(x-5)2-(y-6)29112=1(x5)2(y6)29112=1
Step 8.5.3
Rewrite 112112 as 1111.
Tap for more steps...
Step 8.5.3.1
Use nax=axnnax=axn to rewrite 1111 as 11121112.
(x-5)2-(y-6)29(1112)2=1(x5)2(y6)29(1112)2=1
Step 8.5.3.2
Apply the power rule and multiply exponents, (am)n=amn(am)n=amn.
(x-5)2-(y-6)2911122=1(x5)2(y6)2911122=1
Step 8.5.3.3
Combine 1212 and 22.
(x-5)2-(y-6)291122=1(x5)2(y6)291122=1
Step 8.5.3.4
Cancel the common factor of 22.
Tap for more steps...
Step 8.5.3.4.1
Cancel the common factor.
(x-5)2-(y-6)291122=1
Step 8.5.3.4.2
Rewrite the expression.
(x-5)2-(y-6)2911=1
(x-5)2-(y-6)2911=1
Step 8.5.3.5
Evaluate the exponent.
(x-5)2-(y-6)2911=1
(x-5)2-(y-6)2911=1
(x-5)2-(y-6)2911=1
Step 8.6
Multiply 9 by 11.
(x-5)2-(y-6)299=1
(x-5)2-(y-6)299=1
Step 9
image of graph
(5,6),(4,6),(-5,6)
(
(
)
)
|
|
[
[
]
]
7
7
8
8
9
9
4
4
5
5
6
6
/
/
^
^
×
×
>
>
1
1
2
2
3
3
-
-
+
+
÷
÷
<
<
π
π
,
,
0
0
.
.
%
%
=
=
 [x2  12  π  xdx ]