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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y2=52x
Evaluate
x=52yy
Multiply the terms
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Multiply the terms
52yy
Multiply the terms
52y×y
Multiply the terms
52y2
x=52y2
Rewrite the expression
x=521y2
Swap the sides of the equation
521y2=x
Multiply both sides of the equation by 52
521y2×52=x×52
Multiply the terms
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Evaluate
521y2×52
Multiply the numbers
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Evaluate
521×52
Reduce the numbers
1×1
Simplify
1
y2
y2=x×52
Solution
y2=52x
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Solve the equation
Solve for x
Solve for y
x=52y2
Evaluate
x=52yy
Solution
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Multiply the terms
52yy
Multiply the terms
52y×y
Multiply the terms
52y2
x=52y2
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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
x=52yy
Simplify the expression
x=52y2
To test if the graph of x=52yy is symmetry with respect to the origin,substitute -x for x and -y for y
−x=52(−y)2
Evaluate
−x=52y2
Solution
Not symmetry with respect to the origin
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Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=y26
Calculate
x=52yy
Simplify the expression
x=52y2
Take the derivative of both sides
dxd(x)=dxd(52y2)
Use dxdxn=nxn−1 to find derivative
1=dxd(52y2)
Calculate the derivative
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Evaluate
dxd(52y2)
Rewrite the expression
52dxd(y2)
Evaluate the derivative
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Evaluate
dxd(y2)
Use differentiation rules
dyd(y2)×dxdy
Use dxdxn=nxn−1 to find derivative
2ydxdy
522ydxdy
Calculate
26ydxdy
1=26ydxdy
Swap the sides of the equation
26ydxdy=1
Cross multiply
ydxdy=26
Divide both sides
yydxdy=y26
Solution
dxdy=y26
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Find the second derivative
Find the second derivative with respect to x
Find the second derivative with respect to y
dx2d2y=−y3676
Calculate
x=52yy
Simplify the expression
x=52y2
Take the derivative of both sides
dxd(x)=dxd(52y2)
Use dxdxn=nxn−1 to find derivative
1=dxd(52y2)
Calculate the derivative
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Evaluate
dxd(52y2)
Rewrite the expression
52dxd(y2)
Evaluate the derivative
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Evaluate
dxd(y2)
Use differentiation rules
dyd(y2)×dxdy
Use dxdxn=nxn−1 to find derivative
2ydxdy
522ydxdy
Calculate
26ydxdy
1=26ydxdy
Swap the sides of the equation
26ydxdy=1
Cross multiply
ydxdy=26
Divide both sides
yydxdy=y26
Divide the numbers
dxdy=y26
Take the derivative of both sides
dxd(dxdy)=dxd(y26)
Calculate the derivative
dx2d2y=dxd(y26)
Use differentiation rules
dx2d2y=26×dxd(y1)
Rewrite the expression in exponential form
dx2d2y=26×dxd(y−1)
Calculate the derivative
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Evaluate
dxd(y−1)
Use differentiation rules
dyd(y−1)×dxdy
Use dxdxn=nxn−1 to find derivative
−y−2dxdy
dx2d2y=26(−y−2dxdy)
Rewrite the expression
dx2d2y=26(−y2dxdy)
Calculate
dx2d2y=−y226dxdy
Use equation dxdy=y26 to substitute
dx2d2y=−y226×y26
Solution
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Calculate
−y226×y26
Multiply the terms
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Multiply the terms
26×y26
Multiply the terms
y26×26
Multiply the terms
y676
−y2y676
Divide the terms
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Evaluate
y2y676
Multiply by the reciprocal
y676×y21
Multiply the terms
y×y2676
Multiply the terms
y3676
−y3676
dx2d2y=−y3676
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Rewrite the equation
r=0r=52cos(θ)csc2(θ)
Evaluate
x=52yy
Evaluate
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Evaluate
52yy
Multiply the terms
52y×y
Multiply the terms
52y2
x=52y2
Multiply both sides of the equation by LCD
x×52=52y2×52
Use the commutative property to reorder the terms
52x=52y2×52
Simplify the equation
52x=y2
Move the expression to the left side
52x−y2=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
52cos(θ)×r−(sin(θ)×r)2=0
Factor the expression
−sin2(θ)×r2+52cos(θ)×r=0
Factor the expression
r(−sin2(θ)×r+52cos(θ))=0
When the product of factors equals 0,at least one factor is 0
r=0−sin2(θ)×r+52cos(θ)=0
Solution
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Factor the expression
−sin2(θ)×r+52cos(θ)=0
Subtract the terms
−sin2(θ)×r+52cos(θ)−52cos(θ)=0−52cos(θ)
Evaluate
−sin2(θ)×r=−52cos(θ)
Divide the terms
r=sin2(θ)52cos(θ)
Simplify the expression
r=52cos(θ)csc2(θ)
r=0r=52cos(θ)csc2(θ)
Show Solution
