Answer:
3x +4y = 25
Explanation:
You want the equation of the tangent to the graph of x² +y² = 25 at the point (3, 4).
Tangent
The given equation is that of a circle of radius 5 centered at the origin. For some point (x, y) on the circle, the radius to that point will have a slope of m = y/x. This means the tangent will have a slope of ...
-1/m = -1/(y/x) = -x/y
For the given point, the slope of the tangent is -3/4.
Intercept
The y-intercept in the slope-intercept form equation for a line is 'b'. We can solve for that ...
y = mx +b
b = y - mx
For (x, y) = (3, 4) and m = -3/4, the y-intercept is ...
b = 4 -(-3/4)·3 = 4 +9/4 = 25/4
Slope-intercept equation
Using the values of m and b we found, the slope-intercept form equation for the tangent is ...
y = -3/4x +25/4
Standard form
In standard form, the equation becomes ...
4y = -3x +25
3x +4y = 25
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Additional comment
If you work through this process for point (a, b) on the circle of radius r centered at the origin, you find the equation of the tangent at that point is ...
ax +by = r²
If the center of the circle is at (h, k), then the equation will be the same thing translated to the new center:
(a -h)(x -h) +(b -k)(y -k) = r²
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