Question
Solve the equation
Solve for x
Solve for y
x=2y5
Evaluate
4x×8y=80
Multiply the terms
32xy=80
Rewrite the expression
32yx=80
Divide both sides
32y32yx=32y80
Divide the numbers
x=32y80
Solution
x=2y5
Show Solution

Testing for symmetry
Testing for symmetry about the origin
Testing for symmetry about the x-axis
Testing for symmetry about the y-axis
Symmetry with respect to the origin
Evaluate
4x×8y=80
Multiply the terms
32xy=80
To test if the graph of 32xy=80 is symmetry with respect to the origin,substitute -x for x and -y for y
32(−x)(−y)=80
Evaluate
32xy=80
Solution
Symmetry with respect to the origin
Show Solution

Rewrite the equation
r=∣sin(2θ)∣5sin(2θ)r=−∣sin(2θ)∣5sin(2θ)
Evaluate
4x×8y=80
Evaluate
32xy=80
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
32cos(θ)×rsin(θ)×r=80
Factor the expression
32cos(θ)sin(θ)×r2=80
Simplify the expression
16sin(2θ)×r2=80
Divide the terms
r2=sin(2θ)5
Evaluate the power
r=±sin(2θ)5
Simplify the expression
More Steps

Evaluate
sin(2θ)5
To take a root of a fraction,take the root of the numerator and denominator separately
sin(2θ)5
Multiply by the Conjugate
sin(2θ)×sin(2θ)5×sin(2θ)
Calculate
∣sin(2θ)∣5×sin(2θ)
The product of roots with the same index is equal to the root of the product
∣sin(2θ)∣5sin(2θ)
r=±∣sin(2θ)∣5sin(2θ)
Solution
r=∣sin(2θ)∣5sin(2θ)r=−∣sin(2θ)∣5sin(2θ)
Show Solution

Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=−xy
Calculate
4x8y=80
Simplify the expression
32xy=80
Take the derivative of both sides
dxd(32xy)=dxd(80)
Calculate the derivative
More Steps

Evaluate
dxd(32xy)
Use differentiation rules
dxd(32x)×y+32x×dxd(y)
Evaluate the derivative
More Steps

Evaluate
dxd(32x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
32×dxd(x)
Use dxdxn=nxn−1 to find derivative
32×1
Any expression multiplied by 1 remains the same
32
32y+32x×dxd(y)
Evaluate the derivative
More Steps

Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
32y+32xdxdy
32y+32xdxdy=dxd(80)
Calculate the derivative
32y+32xdxdy=0
Move the expression to the right-hand side and change its sign
32xdxdy=0−32y
Removing 0 doesn't change the value,so remove it from the expression
32xdxdy=−32y
Divide both sides
32x32xdxdy=32x−32y
Divide the numbers
dxdy=32x−32y
Solution
More Steps

Evaluate
32x−32y
Cancel out the common factor 32
x−y
Use b−a=−ba=−ba to rewrite the fraction
−xy
dxdy=−xy
Show Solution

Find the second derivative
Find the second derivative with respect to x
Find the second derivative with respect to y
dx2d2y=x22y
Calculate
4x8y=80
Simplify the expression
32xy=80
Take the derivative of both sides
dxd(32xy)=dxd(80)
Calculate the derivative
More Steps

Evaluate
dxd(32xy)
Use differentiation rules
dxd(32x)×y+32x×dxd(y)
Evaluate the derivative
More Steps

Evaluate
dxd(32x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
32×dxd(x)
Use dxdxn=nxn−1 to find derivative
32×1
Any expression multiplied by 1 remains the same
32
32y+32x×dxd(y)
Evaluate the derivative
More Steps

Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
32y+32xdxdy
32y+32xdxdy=dxd(80)
Calculate the derivative
32y+32xdxdy=0
Move the expression to the right-hand side and change its sign
32xdxdy=0−32y
Removing 0 doesn't change the value,so remove it from the expression
32xdxdy=−32y
Divide both sides
32x32xdxdy=32x−32y
Divide the numbers
dxdy=32x−32y
Divide the numbers
More Steps

Evaluate
32x−32y
Cancel out the common factor 32
x−y
Use b−a=−ba=−ba to rewrite the fraction
−xy
dxdy=−xy
Take the derivative of both sides
dxd(dxdy)=dxd(−xy)
Calculate the derivative
dx2d2y=dxd(−xy)
Use differentiation rules
dx2d2y=−x2dxd(y)×x−y×dxd(x)
Calculate the derivative
More Steps

Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
dx2d2y=−x2dxdy×x−y×dxd(x)
Use dxdxn=nxn−1 to find derivative
dx2d2y=−x2dxdy×x−y×1
Use the commutative property to reorder the terms
dx2d2y=−x2xdxdy−y×1
Any expression multiplied by 1 remains the same
dx2d2y=−x2xdxdy−y
Use equation dxdy=−xy to substitute
dx2d2y=−x2x(−xy)−y
Solution
More Steps

Calculate
−x2x(−xy)−y
Multiply the terms
More Steps

Evaluate
x(−xy)
Multiplying or dividing an odd number of negative terms equals a negative
−x×xy
Cancel out the common factor x
−1×y
Multiply the terms
−y
−x2−y−y
Subtract the terms
More Steps

Simplify
−y−y
Collect like terms by calculating the sum or difference of their coefficients
(−1−1)y
Subtract the numbers
−2y
−x2−2y
Divide the terms
−(−x22y)
Calculate
x22y
dx2d2y=x22y
Show Solution

Conic
5(x′)2−5(y′)2=1
Evaluate
4x×8y=80
Move the expression to the left side
4x×8y−80=0
Calculate
32xy−80=0
The coefficients A,B and C of the general equation are A=0,B=32 and C=0
A=0B=32C=0
To find the angle of rotation θ,substitute the values of A,B and C into the formula cot(2θ)=BA−C
cot(2θ)=320−0
Calculate
cot(2θ)=0
Using the unit circle,find the smallest positive angle for which the cotangent is 0
2θ=2π
Calculate
θ=4π
To rotate the axes,use the equation of rotation and substitute 4π for θ
x=x′cos(4π)−y′sin(4π)y=x′sin(4π)+y′cos(4π)
Calculate
x=x′×22−y′sin(4π)y=x′sin(4π)+y′cos(4π)
Calculate
x=x′×22−y′×22y=x′sin(4π)+y′cos(4π)
Calculate
x=x′×22−y′×22y=x′×22+y′cos(4π)
Calculate
x=x′×22−y′×22y=x′×22+y′×22
Substitute x and y into the original equation 32xy−80=0
32(x′×22−y′×22)(x′×22+y′×22)−80=0
Calculate
More Steps

Calculate
32(x′×22−y′×22)(x′×22+y′×22)−80
Use the commutative property to reorder the terms
32(22x′−y′×22)(x′×22+y′×22)−80
Use the commutative property to reorder the terms
32(22x′−22y′)(x′×22+y′×22)−80
Use the commutative property to reorder the terms
32(22x′−22y′)(22x′+y′×22)−80
Use the commutative property to reorder the terms
32(22x′−22y′)(22x′+22y′)−80
Expand the expression
More Steps

Calculate
32(22x′−22y′)(22x′+22y′)
Simplify
(162×x′−162×y′)(22x′+22y′)
Apply the distributive property
162×x′×22x′+162×x′×22y′−162×y′×22x′−162×y′×22y′
Multiply the terms
16(x′)2+162×x′×22y′−162×y′×22x′−162×y′×22y′
Multiply the numbers
16(x′)2+16x′y′−162×y′×22x′−162×y′×22y′
Multiply the numbers
16(x′)2+16x′y′−16y′x′−162×y′×22y′
Multiply the terms
16(x′)2+16x′y′−16y′x′−16(y′)2
Subtract the terms
16(x′)2+0−16(y′)2
Removing 0 doesn't change the value,so remove it from the expression
16(x′)2−16(y′)2
16(x′)2−16(y′)2−80
16(x′)2−16(y′)2−80=0
Move the constant to the right-hand side and change its sign
16(x′)2−16(y′)2=0−(−80)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
16(x′)2−16(y′)2=0+80
Removing 0 doesn't change the value,so remove it from the expression
16(x′)2−16(y′)2=80
Multiply both sides of the equation by 801
(16(x′)2−16(y′)2)×801=80×801
Multiply the terms
More Steps

Evaluate
(16(x′)2−16(y′)2)×801
Use the the distributive property to expand the expression
16(x′)2×801−16(y′)2×801
Multiply the numbers
More Steps

Evaluate
16×801
Reduce the numbers
1×51
Multiply the numbers
51
51(x′)2−16(y′)2×801
Multiply the numbers
More Steps

Evaluate
−16×801
Reduce the numbers
−1×51
Multiply the numbers
−51
51(x′)2−51(y′)2
51(x′)2−51(y′)2=80×801
Multiply the terms
More Steps

Evaluate
80×801
Reduce the numbers
1×1
Simplify
1
51(x′)2−51(y′)2=1
Use a=a11 to transform the expression
5(x′)2−51(y′)2=1
Solution
5(x′)2−5(y′)2=1
Show Solution
