Quadratics having a common tangent line The curves y = x² + ax...

Question

Quadratics having a common tangent line The curves y = x² + ax + b and y = cx − x² have a common tangent line at the point (1,0). Find a, b, and c.

Accepted Answer

Hi everyone. Let's take a look at this practice problem. This problem says 2s, Y is equal to 2 x 2 plus PX plus Q, and Y is equal to RX minus 3X2. Share a common tangent line at the point of 2 minus 4. Determine the values of PQ and R. So this problem wants us to determine the values of PQ and R in the equations for our two parabolas. So we'll start by looking at the equation for the second parabola. We notice that it only contains the variable R in it. And so we can actually use the point that we're given in the problem, the 2 minus 4, to determine the value of R. So setting Y equal to minus 4 and X equal to 2 in the equation for a second parabola, we will have minus 4 is equal to R multiplied by 2. Minus 3 multiplied by 2 squared. So simplifying this expression, we'll have minus 4 is equal to 2 r minus 12, and now we just need to solve for R. So to solve for R, we'll add 12 to both sides, and so we'll have 8 is equal to 2 R. and dividing both sides by 2, we get r is equal to 4. So that's the value of our first variable found. So, next, we need to determine the values of P and Q. And to do this, we're going to make use of the fact that the two tangent lines or the two paras share a common tangent line at the point of 24. So if the two paras share a common tangent line at this point, that means that the slope of that tangent line must be the same for both parabolas. So, here, just recall to determine the slope of the tangent line at a given point on a curve, that we need to take the derivative of that function with respect to X and evaluate it at the given point. So, in this case, we'll have the derivative of Y with respect to X. Evaluated at X equal to 2. And so for the first parabola, this is going to be equal to the derivative with respect to X. Of the quantity of 2 X squared plus PX plus Q. And that has to be evaluated at X equal to 2. So taking the derivative, this is going to be equal to 4 X plus P. And that is going to be evaluated at X equal to 2. And so evaluating at X equal to 2, this is going to be equal to 4, multiplied by 2, plus P. Which is equal to 8 plus P. So now we have to do the same thing, but for the second problem. So, we'll have the derivative of Y with respect to X, evaluated at X equal to 2. And for the second prevalent, this is going to be equal to the derivative with respect to X. Of the quantity, and here we have R, which is a value of 4, so we'll go ahead and substitute that in. So, we'll have 4 X. Minus 3 X 2 in quantity, and that has to be evaluated at X equal to 2. So, taking the derivative, this is going to be equal to. 4 minus 6 X. And that again still has to be evaluated at X equal to 2. And so doing that evaluation, this is going to be equal to 4 minus 6 multiplied by 2. Which, when we evaluate this expression, is equal to -8. Now, in order to have a common tangent line, these two slopes must be equal, so we'll set 8 plus P equal to minus 8 and solve for P. So to solve for P, we just need to subtract 8 from both sides of the equation, so we'll have P is equal to -16. And so that is our second value found. To find the value of Q, we're just going to use the equation for the first parabola, and the point of 2. minus 4. So, substituting in Y equal to -4 and X equal to 2, and P equal to 16 into the first equation for our parabolas, we will have minus 4. is equal to 2 multiplied by 2 squared. Plus minus 16, multiplied by 2. Plus Q. And so we're just going to simplify this expression. So we'll have minus 4. is equal to 8. Minus 32. Plus Q. And so now to solve for Q, we just need to subtract 8 from both sides and add 32 to both sides. In doing so, we'll get Q is equal to 20. And so that is the final value that we needed to find. So just to recap, we have P is equal to -16, Q is equal to 20, and R is equal to 4. So that is going to be the answer for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Quadratics having a common tangent line The curves y = x² + ax + b and y = cx − x² have a common tangent line at the point (1,0). Find a, b, and c.

Key Concepts

Tangent Line

Derivatives

Quadratic Functions

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