Find an equation of the tangent line to the graph at the given point. ? (y-3)^2=4(x-5), (6,5)

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1 Answer
Feb 8, 2018

#y=x-1#

Explanation:

#(y-3)^2=4(x-5)#

Expanding:

#y^2-6y+9=4x-20#

#y^2-6y-4x=-29#

Differentiating implicitly:

#2ydy/dx-6dy/dx-4=0#

Factor:

#dy/dx(2y-6)=4#

#dy/dx=4/(2y-6)=2/(y-3)#

Gradient of tangent line:

We are only able to use the coordinate of #y# i.e. #5#

#2/(5-3)=1#

Slope point form of a line:

#(y_2-y_1)=m(x_2-x_1)#

#(6,5)#

#:.#

#y-5=(x-6)#

#y=x-1#

GRAPH:

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If you need help with implicit differentiation, you can find it here:

https://socratic.org/calculus/basic-differentiation-rules/implicit-differentiation