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March 31st, 2018, 05:59 AM   #1
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[ASK] Tangent of a Circle

One of the tangent line equation of the circle $\displaystyle x^2+y^2+6x-8y+12=0$ at the point whose abscissa 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:
$\displaystyle (-1)^2+y^2+6(-1)+8y+12=0$
$\displaystyle 1+y^2-6+8y+12=0$
$\displaystyle y^2+8y+7=0$
(y + 1) (y + 7) = 0
y = -1 or y = -7
Then what?

Last edited by skipjack; April 8th, 2018 at 06:51 AM.
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March 31st, 2018, 12:06 PM   #2
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Completing the squares, $\displaystyle x^2+ 6x+ 9+ y^2- 8x+ 16+ 12- 9- 16= (x+ 3)^2+ (y- 4)^2- 13= 0$ so this circle has center (-3, 4) and has radius $\displaystyle \sqrt{13}$. If x= -1, yes, $\displaystyle (-1+ 3)^2+ (y- 4)^2- 13= (y- 4)^2- 9= 0$, $\displaystyle y- 4= \pm 3$ so y= 7 or y= 1. The "abscissa" x= -1 crosses the given circle at (-1, 1) and (-1, 7) so there are two such tangent lines and two possible answers- though the question only asks for "one of" them.

The line through the center (-3, 4) to (-1, 1) has slope (1- 4)/(-1- (-3))= -3/2 so the tangent line, which is perpendicular to the radius, has slope 2/3. The line through (-1, 1) with slope 2/3 is y- 1= (2/3)(x- (-1)). y- 1= (2/3)x+ 2/3, or 3y- 3= 2x+ 2 , 2x- 3y= -5.

The line through the center (-3, 4) to (-1, 7) has slope (7- 4)/(-1-(-3))= 3/2. The tangent line has slope -2/3. The line through (-1, 7) with slope -2/3 is y- 7= (-2/3)(x+ 1), 3y- 21= -2x- 3, 2x+ 3y= 18.
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April 3rd, 2018, 05:58 AM   #3
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Once you have substituted in $\displaystyle x=-1$ to get the two points on the circle $\displaystyle (-1,1)$ and $\displaystyle (-1,7)$ and also completed the square to get $\displaystyle (x+3)^2+(y-4)^2=13$, you can find the two tangent lines by writing the circle this way:

$\displaystyle (x+3)(x+3)+(y-4)(y-4)=13$

and substitute in the points:

$\displaystyle (-1+3)(x+3)+(1-4)(y-4)=13$ to get $\displaystyle 2x-3y=-5$

and

$\displaystyle (-1+3)(x+3)+(7-4)(y-4)=13$ to get $\displaystyle 2x+3y=19$
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April 8th, 2018, 06:36 AM   #4
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Yup, thanks.
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April 8th, 2018, 06:58 AM   #5
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Quote:
Originally Posted by Country Boy View Post
. . . 3y- 21= -2x- 3, 2x+ 3y= 18.
That should be 3y - 21 = -2x - 2, 2x + 3y = 19.
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