MHB What Is the Correct Equation for the Tangent of a Circle at a Given Point?

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The discussion centers on finding the correct equation for the tangent line to the circle defined by the equation x^2 + y^2 + 6x - 8y + 12 = 0 at the point where the x-coordinate is -1. The user initially calculates the y-values at this point, yielding y = -1 or y = -7, and then attempts to derive the tangent line equations. They realize a mistake was made in the initial substitution step and seek clarification on the correct approach to find the tangent line. The conversation emphasizes the importance of accurately determining the slopes and using the correct point to derive the tangent line equations. Ultimately, the user acknowledges a typo in their calculations, indicating a need for careful verification of each step.
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One of the tangent line equation of the circle $$x^2+y^2+6x-8y+12=0$$ at the point whose absis is -1 is ...
A. 2x - 3y - 7 = 0
B. 2x - 3y + 7 = 0
C. 2x + 3y - 5 = 0
D. 2x - 3y - 5 = 0
E. 2x - 3y + 5 = 0

By substituting x = -1, I got:
$$(-1)^2+y^2+6(-1)+8y+12=0$$
$$1+y^2-6+8y+12=0$$
$$y^2+8y+7=0$$
(y + 1) (y + 7) = 0
y = -1 or y = -7
Then what?
 
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Find the center $(h,k)$ of the circle and then find the equations of the two lines that pass through $(h,k),(-1,-1)$ and $(h,k),(-1,-7)$. One of those equations should match one of the answer choices.
 
Wait... A tangent shouldn't go through the circle's center, right?
 
That's right...my mistake. Find the slopes of the lines that pass through the points I gave. Call them $m$. Then find the equations of the lines that have slope $-\frac1m$ and pass through $(-1,-1)$ and $(-1,-7)$. That should get you your answer.
 
Let's try... Judging by the circle's equation, the center is (-3, 4) thus the slope you asked is either $$-\frac{5}{2}$$ or $$-\frac{11}{2}$$ and so the slope of the tangent is either $$\frac{2}{5}$$ or $$\frac{2}{11}$$. Trying to substitute the first one I'll get:
$$y-(-1)=\frac{2}{5}(x-(-1))$$
$$y+1=\frac{2}{5}(x+1)$$
5(y + 1) = 2(x + 1)
5y + 5 = 2x + 2
5y - 2x + 3 = 0
For the second one:
$$y-(-7)=\frac{2}{11}(x-(-1))$$
$$y+7=\frac{2}{11}(x+1)$$
11(y + 7) = 2(x + 1)
11y + 77 = 2x + 2
11y - 2x + 73 = 0
Huh? Where did I go wrong? Or did I misinterpret your hint?
 
I don't see any errors in your work.
 
Monoxdifly said:
One of the tangent line equation of the circle $$x^2+y^2+6x-8y+12=0$$ at the point whose absis is -1 is ...
A. 2x - 3y - 7 = 0
B. 2x - 3y + 7 = 0
C. 2x + 3y - 5 = 0
D. 2x - 3y - 5 = 0
E. 2x - 3y + 5 = 0

By substituting x = -1, I got:
$$(-1)^2+y^2+6(-1)+8y+12=0$$
$$1+y^2-6+8y+12=0$$
$$y^2+8y+7=0$$
(y + 1) (y + 7) = 0
y = -1 or y = -7
Then what?

There's an error in the above. Can you spot it?
 
Yes, the typo in my very first step.
 

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