Question
Identify the conic
Find the standard equation of the parabola
Find the vertex of the parabola
Find the focus of the parabola
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(y+103)2=52(x+409)
Evaluate
48x−72y=30y2×4
Multiply the terms
48x−72y=120y2
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
48x−72y−120y2=0
Move the expression to the right-hand side and change its sign
−72y−120y2=0−48x
Removing 0 doesn't change the value,so remove it from the expression
−72y−120y2=−48x
Use the commutative property to reorder the terms
−120y2−72y=−48x
Multiply both sides of the equation by −1201
(−120y2−72y)(−1201)=−48x(−1201)
Multiply the terms
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Evaluate
(−120y2−72y)(−1201)
Use the the distributive property to expand the expression
−120y2(−1201)−72y(−1201)
Multiply the numbers
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Evaluate
−120(−1201)
Multiplying or dividing an even number of negative terms equals a positive
120×1201
Reduce the numbers
1×1
Simplify
1
y2−72y(−1201)
Multiply the numbers
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Evaluate
−72(−1201)
Multiplying or dividing an even number of negative terms equals a positive
72×1201
Reduce the numbers
3×51
Multiply the numbers
53
y2+53y
y2+53y=−48x(−1201)
Multiply the terms
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Evaluate
−48(−1201)
Multiplying or dividing an even number of negative terms equals a positive
48×1201
Reduce the numbers
2×51
Multiply the numbers
52
y2+53y=52x
To complete the square, the same value needs to be added to both sides
y2+53y+1009=52x+1009
Use a2+2ab+b2=(a+b)2 to factor the expression
(y+103)2=52x+1009
Solution
(y+103)2=52(x+409)
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Solve the equation
Solve for x
Solve for y
x=25y2+3y
Evaluate
48x−72y=30y2×4
Multiply the terms
48x−72y=120y2
Move the expression to the right-hand side and change its sign
48x=120y2+72y
Divide both sides
4848x=48120y2+72y
Divide the numbers
x=48120y2+72y
Solution
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Evaluate
48120y2+72y
Rewrite the expression
4824(5y2+3y)
Cancel out the common factor 24
25y2+3y
x=25y2+3y
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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
Not symmetry with respect to the origin
Evaluate
48x−72y=30y24
Simplify the expression
48x−72y=120y2
To test if the graph of 48x−72y=30y24 is symmetry with respect to the origin,substitute -x for x and -y for y
48(−x)−72(−y)=120(−y)2
Evaluate
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Evaluate
48(−x)−72(−y)
Multiply the numbers
−48x−72(−y)
Multiply the numbers
−48x−(−72y)
Rewrite the expression
−48x+72y
−48x+72y=120(−y)2
Evaluate
−48x+72y=120y2
Solution
Not symmetry with respect to the origin
Show Solution

Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=3+10y2
Calculate
48x−72y=30y24
Simplify the expression
48x−72y=120y2
Take the derivative of both sides
dxd(48x−72y)=dxd(120y2)
Calculate the derivative
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Evaluate
dxd(48x−72y)
Use differentiation rules
dxd(48x)+dxd(−72y)
Evaluate the derivative
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Evaluate
dxd(48x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
48×dxd(x)
Use dxdxn=nxn−1 to find derivative
48×1
Any expression multiplied by 1 remains the same
48
48+dxd(−72y)
Evaluate the derivative
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Evaluate
dxd(−72y)
Use differentiation rules
dyd(−72y)×dxdy
Evaluate the derivative
−72dxdy
48−72dxdy
48−72dxdy=dxd(120y2)
Calculate the derivative
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Evaluate
dxd(120y2)
Use differentiation rules
dyd(120y2)×dxdy
Evaluate the derivative
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Evaluate
dyd(120y2)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
120×dyd(y2)
Use dxdxn=nxn−1 to find derivative
120×2y
Multiply the terms
240y
240ydxdy
48−72dxdy=240ydxdy
Move the variable to the left side
48−72dxdy−240ydxdy=0
Collect like terms by calculating the sum or difference of their coefficients
48+(−72−240y)dxdy=0
Move the constant to the right side
(−72−240y)dxdy=0−48
Removing 0 doesn't change the value,so remove it from the expression
(−72−240y)dxdy=−48
Divide both sides
−72−240y(−72−240y)dxdy=−72−240y−48
Divide the numbers
dxdy=−72−240y−48
Solution
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Evaluate
−72−240y−48
Rewrite the expression
−24(3+10y)−48
Cancel out the common factor −24
3+10y2
dxdy=3+10y2
Show Solution

Find the second derivative
Find the second derivative with respect to x
Find the second derivative with respect to y
dx2d2y=−27+270y+900y2+1000y340
Calculate
48x−72y=30y24
Simplify the expression
48x−72y=120y2
Take the derivative of both sides
dxd(48x−72y)=dxd(120y2)
Calculate the derivative
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Evaluate
dxd(48x−72y)
Use differentiation rules
dxd(48x)+dxd(−72y)
Evaluate the derivative
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Evaluate
dxd(48x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
48×dxd(x)
Use dxdxn=nxn−1 to find derivative
48×1
Any expression multiplied by 1 remains the same
48
48+dxd(−72y)
Evaluate the derivative
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Evaluate
dxd(−72y)
Use differentiation rules
dyd(−72y)×dxdy
Evaluate the derivative
−72dxdy
48−72dxdy
48−72dxdy=dxd(120y2)
Calculate the derivative
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Evaluate
dxd(120y2)
Use differentiation rules
dyd(120y2)×dxdy
Evaluate the derivative
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Evaluate
dyd(120y2)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
120×dyd(y2)
Use dxdxn=nxn−1 to find derivative
120×2y
Multiply the terms
240y
240ydxdy
48−72dxdy=240ydxdy
Move the variable to the left side
48−72dxdy−240ydxdy=0
Collect like terms by calculating the sum or difference of their coefficients
48+(−72−240y)dxdy=0
Move the constant to the right side
(−72−240y)dxdy=0−48
Removing 0 doesn't change the value,so remove it from the expression
(−72−240y)dxdy=−48
Divide both sides
−72−240y(−72−240y)dxdy=−72−240y−48
Divide the numbers
dxdy=−72−240y−48
Divide the numbers
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Evaluate
−72−240y−48
Rewrite the expression
−24(3+10y)−48
Cancel out the common factor −24
3+10y2
dxdy=3+10y2
Take the derivative of both sides
dxd(dxdy)=dxd(3+10y2)
Calculate the derivative
dx2d2y=dxd(3+10y2)
Use differentiation rules
dx2d2y=2×dxd(3+10y1)
Rewrite the expression in exponential form
dx2d2y=2×dxd((3+10y)−1)
Calculate the derivative
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Evaluate
dxd((3+10y)−1)
Evaluate the derivative
−(3+10y)−2×dxd(3+10y)
Evaluate the derivative
−(3+10y)−2×10dxdy
Calculate
−10dxdy×(3+10y)−2
dx2d2y=2(−10dxdy×(3+10y)−2)
Rewrite the expression
dx2d2y=2(−(3+10y)210dxdy)
Calculate
dx2d2y=−(3+10y)220dxdy
Use equation dxdy=3+10y2 to substitute
dx2d2y=−(3+10y)220×3+10y2
Solution
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Calculate
−(3+10y)220×3+10y2
Multiply the terms
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Multiply the terms
20×3+10y2
Multiply the terms
3+10y20×2
Multiply the terms
3+10y40
−(3+10y)23+10y40
Divide the terms
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Evaluate
(3+10y)23+10y40
Multiply by the reciprocal
3+10y40×(3+10y)21
Multiply the terms
(3+10y)(3+10y)240
Multiply the terms
(3+10y)340
−(3+10y)340
Expand the expression
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Evaluate
(3+10y)3
Use (a+b)3=a3+3a2b+3ab2+b3 to expand the expression
33+3×32×10y+3×3(10y)2+(10y)3
Calculate
27+270y+900y2+1000y3
−27+270y+900y2+1000y340
dx2d2y=−27+270y+900y2+1000y340
Show Solution

Rewrite the equation
r=0r=52cos(θ)csc2(θ)−3csc(θ)
Evaluate
48x−72y=30y2×4
Evaluate
48x−72y=120y2
Move the expression to the left side
48x−72y−120y2=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
48cos(θ)×r−72sin(θ)×r−120(sin(θ)×r)2=0
Factor the expression
−120sin2(θ)×r2+(48cos(θ)−72sin(θ))r=0
Factor the expression
r(−120sin2(θ)×r+48cos(θ)−72sin(θ))=0
When the product of factors equals 0,at least one factor is 0
r=0−120sin2(θ)×r+48cos(θ)−72sin(θ)=0
Solution
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Factor the expression
−120sin2(θ)×r+48cos(θ)−72sin(θ)=0
Subtract the terms
−120sin2(θ)×r+48cos(θ)−72sin(θ)−(48cos(θ)−72sin(θ))=0−(48cos(θ)−72sin(θ))
Evaluate
−120sin2(θ)×r=−48cos(θ)+72sin(θ)
Divide the terms
r=5sin2(θ)2cos(θ)−3sin(θ)
Simplify the expression
r=52cos(θ)csc2(θ)−3csc(θ)
r=0r=52cos(θ)csc2(θ)−3csc(θ)
Show Solution
