Question
Solve the equation
Solve for x
Solve for y
x=−y11
Evaluate
−17xy=187
Rewrite the expression
−17yx=187
Divide both sides
−17y−17yx=−17y187
Divide the numbers
x=−17y187
Solution
More Steps

Evaluate
−17y187
Cancel out the common factor 17
−y11
Use b−a=−ba=−ba to rewrite the fraction
−y11
x=−y11
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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
−17xy=187
To test if the graph of −17xy=187 is symmetry with respect to the origin,substitute -x for x and -y for y
−17(−x)(−y)=187
Evaluate
−17xy=187
Solution
Symmetry with respect to the origin
Show Solution

Rewrite the equation
r=−22csc(2θ)r=−−22csc(2θ)
Evaluate
−17xy=187
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
−17cos(θ)×rsin(θ)×r=187
Factor the expression
−17cos(θ)sin(θ)×r2=187
Simplify the expression
−217sin(2θ)×r2=187
Divide the terms
r2=−sin(2θ)22
Simplify the expression
r2=−22csc(2θ)
Evaluate the power
r=±−22csc(2θ)
Solution
r=−22csc(2θ)r=−−22csc(2θ)
Show Solution

Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=−xy
Calculate
−17xy=187
Take the derivative of both sides
dxd(−17xy)=dxd(187)
Calculate the derivative
More Steps

Evaluate
dxd(−17xy)
Use differentiation rules
dxd(−17x)×y−17x×dxd(y)
Evaluate the derivative
More Steps

Evaluate
dxd(−17x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
−17×dxd(x)
Use dxdxn=nxn−1 to find derivative
−17×1
Any expression multiplied by 1 remains the same
−17
−17y−17x×dxd(y)
Evaluate the derivative
More Steps

Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
−17y−17xdxdy
−17y−17xdxdy=dxd(187)
Calculate the derivative
−17y−17xdxdy=0
Move the expression to the right-hand side and change its sign
−17xdxdy=0+17y
Add the terms
−17xdxdy=17y
Divide both sides
−17x−17xdxdy=−17x17y
Divide the numbers
dxdy=−17x17y
Solution
More Steps

Evaluate
−17x17y
Cancel out the common factor 17
−xy
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
−17xy=187
Take the derivative of both sides
dxd(−17xy)=dxd(187)
Calculate the derivative
More Steps

Evaluate
dxd(−17xy)
Use differentiation rules
dxd(−17x)×y−17x×dxd(y)
Evaluate the derivative
More Steps

Evaluate
dxd(−17x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
−17×dxd(x)
Use dxdxn=nxn−1 to find derivative
−17×1
Any expression multiplied by 1 remains the same
−17
−17y−17x×dxd(y)
Evaluate the derivative
More Steps

Evaluate
dxd(y)
Use differentiation rules
dyd(y)×dxdy
Use dxdxn=nxn−1 to find derivative
dxdy
−17y−17xdxdy
−17y−17xdxdy=dxd(187)
Calculate the derivative
−17y−17xdxdy=0
Move the expression to the right-hand side and change its sign
−17xdxdy=0+17y
Add the terms
−17xdxdy=17y
Divide both sides
−17x−17xdxdy=−17x17y
Divide the numbers
dxdy=−17x17y
Divide the numbers
More Steps

Evaluate
−17x17y
Cancel out the common factor 17
−xy
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
22(y′)2−22(x′)2=1
Evaluate
−17xy=187
Move the expression to the left side
−17xy−187=0
The coefficients A,B and C of the general equation are A=0,B=−17 and C=0
A=0B=−17C=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)=−170−0
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 −17xy−187=0
−17(x′×22−y′×22)(x′×22+y′×22)−187=0
Calculate
More Steps

Calculate
−17(x′×22−y′×22)(x′×22+y′×22)−187
Use the commutative property to reorder the terms
−17(22x′−y′×22)(x′×22+y′×22)−187
Use the commutative property to reorder the terms
−17(22x′−22y′)(x′×22+y′×22)−187
Use the commutative property to reorder the terms
−17(22x′−22y′)(22x′+y′×22)−187
Use the commutative property to reorder the terms
−17(22x′−22y′)(22x′+22y′)−187
Expand the expression
More Steps

Calculate
−17(22x′−22y′)(22x′+22y′)
Simplify
(−2172x′+2172y′)(22x′+22y′)
Apply the distributive property
−2172x′×22x′−2172x′×22y′+2172y′×22x′+2172y′×22y′
Multiply the terms
−217(x′)2−2172x′×22y′+2172y′×22x′+2172y′×22y′
Multiply the numbers
−217(x′)2−217x′y′+2172y′×22x′+2172y′×22y′
Multiply the numbers
−217(x′)2−217x′y′+217y′x′+2172y′×22y′
Multiply the terms
−217(x′)2−217x′y′+217y′x′+217(y′)2
Add the terms
−217(x′)2+0+217(y′)2
Removing 0 doesn't change the value,so remove it from the expression
−217(x′)2+217(y′)2
−217(x′)2+217(y′)2−187
−217(x′)2+217(y′)2−187=0
Move the constant to the right-hand side and change its sign
−217(x′)2+217(y′)2=0−(−187)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
−217(x′)2+217(y′)2=0+187
Removing 0 doesn't change the value,so remove it from the expression
−217(x′)2+217(y′)2=187
Multiply both sides of the equation by 1871
(−217(x′)2+217(y′)2)×1871=187×1871
Multiply the terms
More Steps

Evaluate
(−217(x′)2+217(y′)2)×1871
Use the the distributive property to expand the expression
−217(x′)2×1871+217(y′)2×1871
Multiply the numbers
More Steps

Evaluate
−217×1871
Reduce the numbers
−21×111
To multiply the fractions,multiply the numerators and denominators separately
−2×111
Multiply the numbers
−221
−221(x′)2+217(y′)2×1871
Multiply the numbers
More Steps

Evaluate
217×1871
Reduce the numbers
21×111
To multiply the fractions,multiply the numerators and denominators separately
2×111
Multiply the numbers
221
−221(x′)2+221(y′)2
−221(x′)2+221(y′)2=187×1871
Multiply the terms
More Steps

Evaluate
187×1871
Reduce the numbers
1×1
Simplify
1
−221(x′)2+221(y′)2=1
Use a=a11 to transform the expression
−22(x′)2+221(y′)2=1
Use a=a11 to transform the expression
−22(x′)2+22(y′)2=1
Solution
22(y′)2−22(x′)2=1
Show Solution
