Answer:
- 4, -4, 16, 16, true
- tangent: x - y = 8
- normal: y = -x
Explanation:
You want to show that the point (4, -4) is on the graph of x² +y² = 32, and you want the tangent and normal lines at that point.
Point
The point is on the curve because when x = 4 and y = -4 the resulting statement is 16 +16 = 32, which is a true statement.
(a) Tangent
The tangent to a circle is most easily found by first considering the normal. The normal line will be the line through the point on the circle and the center of the circle. Here, the center is (0, 0). The slope of the normal line is ...
m = y/x = -4/4 = -1
The tangent line's slope is the opposite reciprocal of this:
m = -1/(-1) = 1
The point-slope form of the equation for the tangent line is ...
y -k = m(x -h) . . . . . equation for line with slope m through point (h, k)
y -(-4) = 1(x -4) . . . . . equation for line with slope 1 through point (4, -4)
x -y = 8 . . . . . . . . . . add 4-y to put in standard form
The equation of the tangent line is x - y = 8.
(b) Normal
In the section above, we found the slope of the normal line is -1. We also noted that the normal line goes through the point (0, 0). Then the point-slope form of the normal line is ...
y -0 = -1(x -0)
y = -x
The equation of the normal line is y = -x.
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Additional comment
The normal line can also be written as x + y = 0.
The tangent line can also be written as y = x -8.
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