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Algebra Examples
(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(x−h)2a2−(y−k)2b2=1
Vertical hyperbola equation (y-k)2a2-(x-h)2b2=1(y−k)2a2−(x−h)2b2=1
Step 2
Step 2.1
Use the distance formula to determine the distance between the two points.
Distance=√(x2-x1)2+(y2-y1)2Distance=√(x2−x1)2+(y2−y1)2
Step 2.2
Substitute the actual values of the points into the distance formula.
a=√(4-5)2+(6-6)2a=√(4−5)2+(6−6)2
Step 2.3
Simplify.
Step 2.3.1
Subtract 55 from 44.
a=√(-1)2+(6-6)2a=√(−1)2+(6−6)2
Step 2.3.2
Raise -1−1 to the power of 22.
a=√1+(6-6)2a=√1+(6−6)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
Step 3.1
Use the distance formula to determine the distance between the two points.
Distance=√(x2-x1)2+(y2-y1)2Distance=√(x2−x1)2+(y2−y1)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+(6−6)2
Step 3.3
Simplify.
Step 3.3.1
Subtract 55 from -5−5.
c=√(-10)2+(6-6)2c=√(−10)2+(6−6)2
Step 3.3.2
Raise -10−10 to the power of 22.
c=√100+(6-6)2c=√100+(6−6)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
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.
Step 4.4.1
Subtract 11 from both sides of the equation.
b2=100-1b2=100−1
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.
Step 4.6.1
Rewrite 9999 as 32⋅1132⋅11.
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=±√32⋅11b=±√32⋅11
b=±√32⋅11b=±√32⋅11
Step 4.6.2
Pull terms out from under the radical.
b=±3√11b=±3√11
b=±3√11b=±3√11
Step 4.7
The complete solution is the result of both the positive and negative portions of the solution.
Step 4.7.1
First, use the positive value of the ±± to find the first solution.
b=3√11b=3√11
Step 4.7.2
Next, use the negative value of the ±± to find the second solution.
b=-3√11b=−3√11
Step 4.7.3
The complete solution is the result of both the positive and negative portions of the solution.
b=3√11,-3√11b=3√11,−3√11
b=3√11,-3√11b=3√11,−3√11
b=3√11,-3√11b=3√11,−3√11
Step 5
bb is a distance, which means it should be a positive number.
b=3√11b=3√11
Step 6
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=y2−y1x2−x1
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.
Step 6.4.1
Simplify the numerator.
Step 6.4.1.1
Multiply -1−1 by 66.
m=6-65-(-5)m=6−65−(−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.
Step 6.4.2.1
Multiply -1−1 by -5−5.
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(x−h)2a2−(y−k)2b2=1.
(x-h)2a2-(y-k)2b2=1(x−h)2a2−(y−k)2b2=1
(x-h)2a2-(y-k)2b2=1(x−h)2a2−(y−k)2b2=1
Step 7
Substitute the values h=5h=5, k=6k=6, a=1a=1, and b=3√11b=3√11 into (x-h)2a2-(y-k)2b2=1(x−h)2a2−(y−k)2b2=1 to get the hyperbola equation (x-(5))2(1)2-(y-(6))2(3√11)2=1(x−(5))2(1)2−(y−(6))2(3√11)2=1.
(x-(5))2(1)2-(y-(6))2(3√11)2=1(x−(5))2(1)2−(y−(6))2(3√11)2=1
Step 8
Step 8.1
Multiply -1−1 by 55.
(x-5)212-(y-(6))2(3√11)2=1(x−5)212−(y−(6))2(3√11)2=1
Step 8.2
One to any power is one.
(x-5)21-(y-(6))2(3√11)2=1(x−5)21−(y−(6))2(3√11)2=1
Step 8.3
Divide (x-5)2(x−5)2 by 11.
(x-5)2-(y-(6))2(3√11)2=1(x−5)2−(y−(6))2(3√11)2=1
Step 8.4
Multiply -1−1 by 66.
(x-5)2-(y-6)2(3√11)2=1(x−5)2−(y−6)2(3√11)2=1
Step 8.5
Simplify the denominator.
Step 8.5.1
Apply the product rule to 3√113√11.
(x-5)2-(y-6)232√112=1(x−5)2−(y−6)232√112=1
Step 8.5.2
Raise 33 to the power of 22.
(x-5)2-(y-6)29√112=1(x−5)2−(y−6)29√112=1
Step 8.5.3
Rewrite √112√112 as 1111.
Step 8.5.3.1
Use n√ax=axnn√ax=axn to rewrite √11√11 as 11121112.
(x-5)2-(y-6)29(1112)2=1(x−5)2−(y−6)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)29⋅1112⋅2=1(x−5)2−(y−6)29⋅1112⋅2=1
Step 8.5.3.3
Combine 1212 and 22.
(x-5)2-(y-6)29⋅1122=1(x−5)2−(y−6)29⋅1122=1
Step 8.5.3.4
Cancel the common factor of 22.
Step 8.5.3.4.1
Cancel the common factor.
(x-5)2-(y-6)29⋅1122=1
Step 8.5.3.4.2
Rewrite the expression.
(x-5)2-(y-6)29⋅11=1
(x-5)2-(y-6)29⋅11=1
Step 8.5.3.5
Evaluate the exponent.
(x-5)2-(y-6)29⋅11=1
(x-5)2-(y-6)29⋅11=1
(x-5)2-(y-6)29⋅11=1
Step 8.6
Multiply 9 by 11.
(x-5)2-(y-6)299=1
(x-5)2-(y-6)299=1
Step 9
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