Theory and Examples

Tangents Find equations f...

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Theory and Examples

Tangents Find equations for the tangents to the circle (x − 2)² + (y − 1)² = 5 at the points where the circle crosses the coordinate axes.

Accepted Answer

Hello there, today we are 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. Determine the equations of the tangent lines to the circles of x minus 22 plus y minus 42 is equal to 13 at every point where the curve intersects the x and y axes. OK. So it appears for this particular prompt based on all the information that is provided to us we're asked to ultimately determine what the equations of the tangent lines are to this specific circle that is given to us by the prom itself. So we're ultimately asked to solve for. A couple different answers, since we're trying to determine what equations, which means more than one, so we're trying to find the equations of this tangent line, the equations of the tangent lines to this specific circle. So with that in mind, now that we know what we're ultimately trying to solve for, based on our prior knowledge, in order to solve for this particular problem, we will first need to identify the points where the circle crosses the axes. These will be the intercepts, and then we will need to find the slope of the circle at these specific points, and we will do this by using implicit differentiation. Then we will need to recall and use the point slope form in order to write the equations of the tangent lines themselves. So that will be like our bro, like, once again, this is overall what our blueprint will be, so this is the blueprint, our a basic idea of what we're gonna do to solve this problem. So our first step that we need to use. we need to find our intersection points, which will be our intercepts. So for the Y intercepts, this will be where X is equal to 0, that means we need to substitute X is equal to 0 into our circle equation. So we can go ahead and write 0 minus 2 to 2. Plus Y minus 4 to 2 is equal to 13. So essentially what we need to do is we need to use a little bit of algebra to rearrange this expression. To start to solve for why. On one side of our expression, so we can note that this expression, when we rearrange it, we can write that Y minus 4 will be equal to plus or minus 3. Which will give us Y is equal to 7. And Y is equal to 1. So once again, like I mentioned earlier, we need to rearrange our expression to isolate and solve for Y, and when we do, we will find that Y is equal to 7 and 1, so that will mean that our points are parentheses 0.7 and parentheses of 0.1. And in order to solve for the X intercepts, this will be where Y is equal to 0, so we plug in a value of 0 in for Y into our circle equation. So we'll have parentheses X minus 22 + 0 minus 42 is equal to 13. Which will give us that parentheses of x minus 22 is equal to -3. And we need to note that since a squared real number cannot be negative, that will mean as a result, there will be no X intercepts as a result. So for our second step now, we need to find the derivative which will be the slope of the tangent. So at this point we will recall and use implicit differentiation on parentheses of x minus 22 plus parentheses of y minus 4 to 2, which will be equal to 13. And this will be with respect to X. So, with that said. We can go ahead and write 2 multiplied by parentheses of X minus 2 + 2 multiplied by parentheses of Y minus 4 multiplied by DY by DX, which will be equal to 0. So solving for DY by DX, we can write 2 multiplied by parentheses of Y minus 4. Multiplied by DY by DX is equal to -2 multiplied by parentheses of X minus 2. Fantastic. So that will mean as a result that DY by DX will be simply equal to X minus 2 divided by Y minus 4. Fantastic. So for our third and final step, we need to determine the tangent equations at our intercepts, and we need to note that we only have two points to consider, and those are once again 0.7 and 0.1. So, starting with 0.0.7, we have our slope. Which we'll call M1 and M1 will be equal to 0 minus 2 divided by 7 minus 4. Which will be equal to 2 divided by 3. And once again, we're using our 0.0.7. So we're plugging in a value of 0 in for X and a value of 7 in for Y. So now we need to use our equation. So using our point slope form, we'll have Y minus Y1 is equal to m multiplied by X minus X1. Which will mean that y minus 7 will be equal to 2 divided by 3 multiplied by x minus 0, which when we rearrange this expression to isolate and solve for Y, we will find that y is simply equal to 2 divided by 3 multiplied by x plus 7. Now focusing our attention on the 0.0.1, that will mean that M2 will be equal to 0 minus 2 divided by 1 minus 4, which will be equal to -2 divided by 3, which will mean that y minus 1 will be equal to -2 divided by 3 multiplied by parentheses of x minus 0. Which will mean that y will be equal to -2 divided by 3 x plus 1, and that's it. These are our final two answers for the tangent lines. Hooray, we did it. So going back to the top to recap what we've learned, we could safely say that our final answers for the equations of the tangent lines will be as follows. So our final two answers once again are Y is equal to 2 divided by 3 X plus 7, and Y is equal to -2 divided by 3 X + 1. And that's it. Those are our final two answers. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Theory and ExamplesTangents Find equations for the tangents to the circle (x − 2)² + (y − 1)² = 5 at the points where the circle crosses the coordinate axes.

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Theory and Examples

Tangents Find equations for the tangents to the circle (x − 2)² + (y − 1)² = 5 at the points where the circle crosses the coordinate axes.