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?

<IMAGE>

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Consider the parabola. X is equal to -2 to the power of 2. From the point parentheses C.0. It is possible to draw three normal lines to the curve. One of these normal lines is the X axis. Determine the value of C so that the other two normal lines are perpendicular to each other. Awesome. So it appears for this particular problem, we're ultimately trying to determine a value of C, such that the other two normal lines are perpendicular to each other. So perpendicular, meaning that they form a right angle with the normal line. So that's what perpendicular means. So it makes a 90 degree angle. So once again, we're trying to figure out this value of C, such that these two normal lines will be perpendicular to each other. So with that said, let's look at the figure that's provided to us by the prom itself, and as you can see, we have our parabola represented by a red curve for X is equal to -2Y2 or Y to the power 2. And as you can see, we have our three normal lines where one of our normal lines is the X axis, which is represented by a bold black line. Then we have our other two normal lines which are represented by linear curves that are blue and green, that are touching the arms, or you might have heard them called legs of our parabola. And as you can see, we have red points on either side of our legs of our parabola, indicated by red dots with dotted lines going towards the X axis. And as you could see, these dotted lines hit our normal curves or our normal lines at 90 degree angles which are represented by like the 90 degree symbol in orange. Awesome. So as you can see, it is perpendicular at 3 different spots. Awesome. And of course, the vertical axis is our Y axis, and then we have our point C.0 indicated on our coordinate plane for our figure. Fantastic. So now that we have a visualization of what's happening in this particular prompt. Our first step that we need to take is we need to note and recall that the derivative of a function at a given point represents the slope of the tangent line to the function at that point. So with that said, we need to determine the derivative of X equals -2 to the power of 2. So when we determine the derivative, we will find that 1 is equal to -2 multiplied by 2, Y multiplied by DY by DX. Which will give us -4 Y multiplied by DY by DX is equal to 1, fantastic. And when we rearrange this expression to isolate so for DY by DX by itself, you will find that DY by DX is going to be simply equal to -1 divided by 4Y, which we can also state that M tangent, so once again, since we're told that M is the slope of the tangent line, so the slope of the tangent line, Mtan for short, will be equal to -1 divided by 4Y. Awesome. So for the two lines to be perpendicular, as we should recall, the product of their slopes must be -1, meaning that the slope of the tangent line, multiplied by, so the slope of our tangent line multiplied by The slope of our perpendicular. is going to be equal to -1. So that will mean that since we know our value of the slope of the tangent line, so it's gonna be negative 1 divided by 4 Y multiplied by M perpendicular is going to be equal to -1. So rearranging this equation to isolates solve for M perpendicular all by itself, that will mean that M perpendicular is going to be equal to simply 4Y. Awesome. So now at this point we need to let parentheses X 1 Y1 be a point on the curve where the tangent line is perpendicular to the line connecting 0. And X1.com Y1. Fantastic. So with that said, the slope of the line connecting parentheses and parentheses X1, Y1 is M perpendicular, meaning we can go ahead and write that M perpendicular is equal to Y1 minus 0 divided by X1 minus C. So that will mean we could go ahead and write that 4 multiplied by Y1 is equal to Y1 divided by X1 minus C. And we can go ahead and write that X1 minus C is equal to 1/4, and when we rearrange this expression to isolate inlve for C, we will find that C is simply equal to X1 minus 1/4. Fantastic. So now, our next step is we need to solve for Y1 in terms of X1. So X1 is equal to -2Y1 to the power of 2. Which we can simplify to Y 1 squared is equal to negative. X1 divided by, so it's going to be negative X1 divided by 2. So getting Y all by itself, we will find that, or rather Y1 all by itself. So Y1 when we rearrange the equation to isolate sulfur Y1, meaning we have to take the square root of both sides, we will find that Y1 is equal to plus or minus the square root of, so once again, plus or minus the square root of negative X1 divided by 2. Fantastic. So, with that in mind, as we should recall, by symmetry, the two points on the parabola are parentheses X1, the square root of negative X1 divided by 2. And parentheses X1 negative square root of negative X1 divided by 2. Fantastic. So, with that said, so since the normal lines from parentheses C 0 to parentheses X1, the square root of negative X1 divided by 2 and 2. X1 negative square root of negative X1 divided by 2 are perpendicular, we can go ahead and write that parenthesis, the square root of negative X1 divided by 2 minus 0 divided by X1 minus C. And don't forget that this is in parentheses multiplied by parentheses. Negative square root of X1 divided by 2, and don't forget that it's negative X1 divided by 2. So once again, we have our first parenthesis, which is the square root of negative X1 divided by 2 minus 0, all divided by X1 minus C. In parentheses multiplied by negative square root of negative X1 divided by 2. So once again, this is important, negative X1 divided by 2 minus 0, all divided by X1 minus C will be all equal to -1, fantastic. So now, what we need to do is we're going to be rearranging this expression to isolate and solve for X1. So in an effort to save time, we're going to skip several steps, but essentially all we're trying to do is we're taking this big old long expression and we're just rearranging it to isolate and solve for X1. And when we do that, we will find that X1 is going to be simply equal to -18 when we use our algebra to simplify down to its most simple form, and we rearrange everything to get X1 all by itself. So now that we know that X1 is equal to -18, we can now solve for C, which is our final answer that we're ultimately trying to solve for. So let us note that C is equal to -18. Minus 1/4, which is going to be equal to -3/8, and that is our final answer. Hooray, we did it. So going back to look. At our prom itself to quickly recap what we've learned, that will mean. Without a doubt, our final answer has to be. C equals, so C equals -38 and that is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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?

Key Concepts

Normal Line to a Curve

Parabola and its Properties

Perpendicular Lines

Watch next