Witch of Agnesi Let y(x²+4)=8 (see figure).

Question

b. Find equations of all lines tangent to the curve y(x²+4)=8 when y=1.

Accepted Answer

Hello. In this video, we are going to be calculating the following tangent lines. So we are asked to determine the equations of all tangent lines to Y multiplied by X2 + 16 equal to 64, at Y is equal to 2. So, whenever we are asked to find the equation of a tangent line, there are two pieces of information we need. We need a slope M, and we need a corresponding point to the tangent line, X1 Y1. Currently, we are given the Y value for the tangent line's point, but we don't have the corresponding X value. Let's go ahead and start this problem by solving for the X value for the point. So, the equation given to us is Y multiplied by X2 + 16, equal to 64. By taking the value of Y and substituting it into the equation, we can rewrite the equation as 2 multiplied by X2 + 16, equal to 64. Now what we are going to do is we are going to solve for X. We will divide both sides of this equation by 2, leaving us with X2 + 16 equal to 32. We will then subtract 16 to both sides of the equation, leaving us with X2 equal to 16. And by taking the square root of both sides of the equation, we will be left with two values of X. X is equal to positive and negative 4. What this means is that there are two points where we have a tangent line to the given equation. The first point will be located at 4.2, and the second point will be located at -4.2. What this means is there are two tangent line equations that we are going to need to solve for. So, let's go ahead and start by solving for the tangent line equation using the point 4.2. Now, in order to solve for the tangent line equation, we have the points, but now we need to solve for the slope. Normally, when we are asked to find the slope to the equation of a tangent line, we will take the derivative of the given equation. In the previous video, we did solve for this derivative using implicit differentiation. That derivative was DY divided by DX equal to -2 XY divided by the quantity X2 + 16. Since we have the respective X and Y quantities, we can plug in our point to solve for the slope for the tangent line. So, for 4.2, we will take the slope or the derivative DYDX, and plug in the respective X and Y values. That is going to be -2, multiplied by 4, multiplied by 2, divided by 42 + 16. This will simplify to be negative. To be -16. Divided by 32, which will further simplify to be -12. So the slope at the point 4.2 is going to be -12. And now that we have the slope and point for the first tangent line, we can use the point slope form to solve for the equation. That will be Y minus Y1, which is 2, equal to the slope, -12 multiplied by X minus X1, which is 4. We will distribute -12 into X minus 4. That will simplify the equation of the line to be Y minus 2 equal to -12 X plus 2. Then we will add 2 to both sides of the equation, and that will leave us with Y equal to -12 X plus 4. This is going to be the equation of the first tangent line at the point 4.2. Now we want to go ahead and repeat this process, but for the point, -4.2. So again, we need to get the respective slope at this point. In order to find that slope, we will use the derivative D Y D X equal to negative to XY divided by X2 + 16. Then we will go ahead and plug in the respective X and Y values. That will give us -2 multiplied by -4, multiplied by 2, divided by -4 squared plus 16. This will simplify to leave us with 16 divided by 32, which will further simplify to be positive 1/2. This is going to be the respective slope at the point -4.2. And now we can take this value and plug it into the point slope form to solve for the second tangent line. That will be Y minus Y1, which is 2, equal to the slope, 1/2 multiplied by X minus X1, which is -4, so that will be X plus 4. We will distribute 1/2 to X + 4, and that will allow us to rewrite the equation of the line to be Y minus 2 equal to 12 X plus 2. Then we will add 2 to both sides of the equation, and the equation of the second tangent line will be Y is equal to 1/2 X plus 4. And both of these solutions are going to be the solution to the original problem. So, I hope this video helps you in understanding how to find these tangent lines, and I will go ahead and see you all in the next video.

Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>
b. Find equations of all lines tangent to the curve y(x²+4)=8 when y=1.

Key Concepts

Implicit Differentiation

Tangent Line Equation

Witch of Agnesi

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