Normal lines to a parabola Show that if it is...

Question

Normal lines to a parabola Show that if it is possible to draw three normal lines from the point (a, 0) to the parabola x = y² shown in the accompanying diagram, then a must be greater than 1/2. One of the normal lines is the x-axis. For what value of a are the other two normal lines perpendicular?

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Accepted Answer

Welcome back, everyone. Determine the constant C so that the line Y equals -2 divided by 9 X plus C meets the graph of Y2 equals 3x cubed orthogonally given that the point of intersection is in quadrant 1. A 29 divided by 3, B-25 divided by 3, C 29 divided by 3, and D 25 divided by 3. So let's begin with the graph we're given the Y2 equals 3 x cubed. What we're going to do is simply identify the slope of the tangent line to this curve, so we're going to perform an implicit differentiation, differentiating both sides, we get 2 Y multiplied by Y on the left hand side. So we are applying the power rule followed by the chain rule. On the right hand side we can apply the power rule which gives us 3 multiplied by. The derivative of X cubed, which is 3 x 2. So now solving for Y we can show that Y equals 9 x 2, the right hand side divided by 2 Y, the left hand side, right? This is our expression of the slope, and now we can consider the slope of the given line, right? M1 is -2 divided by 9, so we have determined the slope of the line. That means the graph orthogonally, and that means that those slopes must be perpendicular to each other. If we're looking for a slope M2, we have to recall the condition for two slopes being perpendicular to each other. The product of those slopes must be equal to -1. Which means that the slope to our. Tent line Must be equal to -1 divided by M1. Which is -1. Divided by -2 divided by 9. And this gives us nine halves. So we can now equate Y. To 9 halves, because Y represents the slope in a general form. And we know that Y is 9 halves multiplied by X2 divided by Y. So what can we tell? Well, we can say that X2 divided by Y must be equal to 1, right? Because 9 halves multiplied by 1 gives us 9 halves. So X2d must be equal to Y. What we're going to do is simply use this relationship and substitute it into the equation of the curve, right? So we're going to get Y2 equals. We X cubed. This is the original expression, and now if X2 is equal to Y, then Y2 is going to be equal to x to the power of 4. We also know that X and Y, they are both positive because we're looking at quadrant 1, right? So we also have to keep that in mind and now we're going to substitute. X to the power of 4 equals 3 x cubed. What we can do is simply perform factorization. First of all, let's subtract 3x cubed from both sides, which gives us X to the power of 4 minus 3X cubed, and that's X cubed in C X minus 3. So this gives us two possible solutions. X can be equal to 0, or X can be equal to 3. We can exclude 0, right, because it represents the origin and our X should be positive. So we're going to use X equals 3 moving forward. If X equals 3, this means that Y is going to be equal to X2, which is 32 and that's 9. So we have the point of interest, specifically, we are considering the point of tangency, and that point is X equals 3, Y equals 9. And now we can use this point to solve for C. First of all, we can show that C is equal to Y plus 2 divided by 9 X. So we're basically adding 2 divided by 9 X to both sides, and now we can solve for C using the coordinates of that point. So C equals 9 + 2 divided by 9 multiplied by 3. Which is 9+. 6 divided by 9 or. Simply 9 + 2/3. And now finding the common denominator, 9 multiplied by 3 gives us 27. We're adding 2. And our denominator is 3, so we get 29 divided by 3, which corresponds to the answer choice C. Thank you for watching.

Key Concepts

Normal Line

Parabola Properties

Perpendicular Lines

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