Find equations for the tangent line and normal line to the...

Find equations for the tangent line and normal line to the circle at each given point. (The normal line at a point is perpendicular to the tangent line at the point.) Use a graphing utility to graph the equation, the tangent lines, and the normal lines. x2 + y2 = 25 (3, 4), (-4, -3)

Transcript

00:01 In this question, we are given a circle x square plus y square goes 25.

00:07 We want to find the equation of the tangent and normal at this point 3, 4, and also at this point minus 4 minus 3, and to use a graphing utility to graph the circle and these lines.

00:19 Now before we proceed, we need to know that gradient at a point, for instance, at a point p where the x is equals to x -not, it will be equal to the differentiation of the function at the x equals to x not.

00:41 And also we need to know the normal gradient will be minus 1 over the tangent gradient.

00:55 And also the equation of line will be y minus y1 equals to m x minus x1 where m is the gradient and x1 y1 is a point on the line.

01:14 So let's find the differentiation for this.

01:18 So we have this.

01:22 Let's differentiate with respect to x on both sides.

01:34 So for x square, when we differentiate respect to x, we will get 2x plus.

01:42 Now, y square, when i differentiate, bring down the power to, freeze the y, subtract one to the power and there will just be 1.

01:48 And don't forget to differentiate y respect to x, and that will be just d, y, dx, equals to, now 25 is constant, when differentiate respect to x, you get zero.

01:59 So this gives me d, y, dx is minus x over y.

02:03 So at the point 3 ,4, so at 3 ,4, my tangent gradient would be, dy, dx, will be minus 3 over 4...

Question

Find equations for the tangent line and normal line to the circle at each given point. (The normal line at a point is perpendicular to the tangent line at the point.) Use a graphing utility to graph the equation, the tangent lines, and the normal lines. x2 + y2 = 25 (3, 4), (-4, -3)

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