SOLUTION: Write the equation of the tangent line to (x-2)²+(y+2)²=169 at the point (7,10). Please answer as soon as possible. I have a test tomorrow and my teacher has not given enough ti

Algebra ->  Equations -> SOLUTION: Write the equation of the tangent line to (x-2)²+(y+2)²=169 at the point (7,10). Please answer as soon as possible. I have a test tomorrow and my teacher has not given enough ti      Log On


   

Question 923109: Write the equation of the tangent line to (x-2)²+(y+2)²=169 at the point (7,10).
Please answer as soon as possible. I have a test tomorrow and my teacher has not given enough time into teaching us this! Please help! I need to know all the steps and how to get it. I know that it does include y=mx+b and I think the distance and midpoint formulas though.
Found 2 solutions by josgarithmetic, lwsshak3:
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
Find derivative and then evaluate the derivative at the point (7,10).

2%2A%28x-2%29%2A1%2B2%28y%2B2%29%2A%28dy%2Fdx%29=0

x-2%2B%28y%2B2%29%28dy%2Fdx%29=0

%28y%2B2%29%28dy%2Fdx%29=-x%2B2

%28dy%2Fdx%29=%28-x%2B2%29%2F%28y%2B2%29

What is the value of this rate of change at the given point?
dy%2Fdx=%28-7%2B2%29%2F%2810%2B2%29
highlight%28-5%2F12%29.
That is a slope.

You are more completely looking for highlight_green%28y=-%285%2F12%29x%2Bb%29, and you know x and y from the given point, but you do not yet know b.

Answer by lwsshak3(11628) About Me  (Show Source):
You can put this solution on YOUR website!
Write the equation of the tangent line to (x-2)²+(y+2)²=169 at the point (7,10).
***
Given equation is that of a circle with center at (2,-2) and radius=√169=13
The tangent line through the point of tangency(7,10) is perpendicular to a 2nd line through the same point of tangency and center of the circle (2,-2). Therefore, the slope of the tangent line is the negative reciprocal of this 2nd line.
..
slope of 2nd line=∆y/∆x=(10-(-2))/(7-2)=12/5
slope of tangent line=-5/12
equation of tangent line: y=-5x/12+b
solve for b using coordinates of the point(7,10)on the line
10=-5*7/12+b
b=10+35/12=155/12
equation of tangent line: y=-5x/12+155/12