Question 832263
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Hi, again--

THE PROBLEM:
Find the general equation of the circle with its center at the origin and tangent to line 5x + 12y = 26.

A SOLUTION:
We want the equation for the circle in this form.
x^2 + y^2 + Dx + Ey + F = 0

Our first step is to find the point of tangency for the line 5x + 12y = 26. We use the fact that line through 
the center of the circle and the point of tangency (called the normal line) and the tangent line are 
perpendicular. 

Find the slope of the tangent line. (Convert the equation to slope-intercept form.)
5x + 12y = 26
12y = -5x + 26
y = -(5/12)x + 26/12

The slope of the tangent line is -5/12.

Find the slope of the normal line. Recall that the slopes of perpendicular lines are related by 
{{{m[1]*m[2]=-1}}}. The slope of the normal line is 12/5 since (-5/12)(12/5) = -1.

Use the point-slope form to find an equation for the normal line. The slope is 12/5 the point is (0,0) 
since the line passes through the origin.
y - 0 = (12/5)(x-0)

Simplify  to give the equation for the normal line.
y = (12/5)x

Since the normal line and tangent line intersect at the point of tangency, we solve the system of equations 
to find that intersection point.

5x + 12y = 26
y = (12/5)x

Substitute (12/5)x for y in the first equation.

5x + 12(12/5)x = 26
5x + 144x/5 = 26

Multiply every term by 5 to clear the denominator.
25x + 144x = 130
169x = 130
x = 130/169
x = 10/13

Substitute 10/13 for x in the second equation.
y = (12/5)x
y = (12/5)(10/13)
y = 24/13

The normal line and the tangent line intersect at the point (10/13, 24/13). More important, this point is 
on the circle. The distance between this point and origin is the radius of the circle. Find the radius r 
using the distance formula.

{{{r=sqrt((10/13-0)^2+(24/13-0)^2)}}}
{{{r=sqrt(100/169+576/169)}}}
{{{r=26/13}}}

An equation for a circle with radius r and center (h,k) is
{{{(x-h)^2+(y-k)^2=r^2}}}

Substitute 0 for h, 0 for k and 26/13 for r.
{{{(x-0)^2+(y-0)^2=(26/13)^2}}}

Simplify and rewrite the equation in general form.
{{{x^2+y^2-676/169=0}}}



Hope this helps! Feel free to email if you have any questions about the solution.

Good luck with your math,

Mrs. F
math.in.the.vortex@gmail.com
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