How do I solve this Pre-Calc problem on tangent lines?

Question

How do I write the equation of the tangent to the curve through point P in standard form  for the equation  (x-2)^2+(y-3)^2=25?

Answer 1

Find y'(x,y) and evaluate it at point P(x₁,y₁).  That gives you the slope of the tangent line at P.  Call it m.  Once you have m, the equation of the line in point-slope form is

 

y - y₁ = m(x - x₁)

 

You can easily convert this to standard form.  To get y'(x,y), use implicit differentiation.

 

2(x - 2) + 2(y - 3)y' = 0

 

x - 2 + (y - 3)y' = 0

 

y'(x,y) = -(x - 2) / (y - 3)

 

m = y'(x₁,y₁)

 

Proceed to write the point-slope form of the line, and then convert to standard form.