131k views
4 votes
Verify that the given point is on the curve and find the lines that are a. tangent and b. normal to the curve at the given point.

Verify that the given point is on the curve and find the lines that are a. tangent-example-1

1 Answer

4 votes

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.

__

Additional comment

The normal line can also be written as x + y = 0.

The tangent line can also be written as y = x -8.

<95141404393>

Verify that the given point is on the curve and find the lines that are a. tangent-example-1
User Eku
by
8.7k points

No related questions found