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If two points $A(a, 0)$ and $B(-a, 0)$ are stationary and if $angle A-angle B=heta$ in $Delta A B C$, the locus of $C$ is(A) $x^{2}+y^{2}+2 x y an heta=a^{2}$(B) $x...

Question

If two points $A(a, 0)$ and $B(-a, 0)$ are stationary and if $angle A-angle B=heta$ in $Delta A B C$, the locus of $C$ is(A) $x^{2}+y^{2}+2 x y an heta=a^{2}$(B) $x^{2}-y^{2}+2 x y an heta=a^{2}$(C) $x^{2}+y^{2}+2 x y cot heta=a^{2}$(D) $x^{2}-y^{2}+2 x y cot heta=a^{2}$

If two points $A(a, 0)$ and $B(-a, 0)$ are stationary and if $angle A-angle B= heta$ in $Delta A B C$, the locus of $C$ is (A) $x^{2}+y^{2}+2 x y an heta=a^{2}$ (B) $x^{2}-y^{2}+2 x y an heta=a^{2}$ (C) $x^{2}+y^{2}+2 x y cot heta=a^{2}$ (D) $x^{2}-y^{2}+2 x y cot heta=a^{2}$



Answers

Show that the parametrization of the ellipse by the angle $\theta$ is
$$\begin{aligned} x &=\frac{a b \cos \theta}{\sqrt{a^{2} \sin ^{2} \theta+b^{2} \cos ^{2} \theta}} \\ y &=\frac{a b \sin \theta}{\sqrt{a^{2} \sin ^{2} \theta+b^{2} \cos ^{2} \theta}} \end{aligned}$$

We want to show that the parameter ization of lips by an angle. Actually, angular cornet data is given by these two lengthy expressions. So this is a much, much the same parameter ization of lips but McDuck clear form than the one in the previous problem. Although in this case, data is actually the angle. So this is actually a problem. A very ugly problem with a ton of Alex Birch. Now try to walk you through it. So now, while we're looking at this angle data so that that measures the angle between whatever this line is too the point on the ellipse and the X axis. So we have our point, Ex general point X Y and we know that this this will call this distance here. L Okay, so the length from the origin to the point on the lift. Now we know from before that we can say X equals a co Santee. And why would be society as simple privatization of the lives? We also know that why l squared equals x squared plus y squared. Thank you. And then we can say that, um, sign of data or else Sinus data is why so again triggered. So l sign of data is this distance So that's why. And so then we know that sign of favor. He goes b sine of t over hell, and we get likewise. We can say l co scientist is X. So we know co signed data is X over? No. Which is a co sign t over. Okay, so that that's simple enough. So far, the problem is, um, we need to get, um l in terms of data. So we know now that we know that X equals l co signed data. Why? Because l scientist before so x squared equals again when we got trig functions like this a lot. It's a lot of times. It's a good idea to square things. So X squared equals l square co sign squared data and then we can plug that in to our and we just get that. We know that l could get elsewhere from before we know l squared is X squared plus y squared. And likewise, we know that. Why ISS Why square is l squared sine squared data and we know elsewhere from before from here. So we got why square and we got X squared Now we need to we need to eliminate Except why we want to solve for X and y X squared Life square here. So what? We also know that from our equation for any lips we have that x squared We know that X cried over a squared plus y squared over b squared equals one So x squared equals 80 square times a quantity Why one minus y squared over B square Hey, so we can take that guy and substitute in there And I would get an expression simply for terms of y squared and science assigned a data squared And we can likewise we could do the same thing. Solve the equation for any lips for why squared So why squared equals B squared one minus expert over a square Take that and substitute in there And now we get an expression for X squared. Now, if we do it all that algebra we wind up with, um why square equals this whole quantity A squared one times one minus y squared over B square. Okay, this was X square plus y squared sine squared. We can manipulate that around a little bit and So we get a square plus y squared times quantity one minus a squared over B square. All times science quite fatal. And we can pull everything, um, kind of manipulate that further and so get all the Y squared on one side. So we get this wife square here, and then we get a negative wife squared sine squared data there and a positive Why squared a squared over B squared sine squared data there and on the right hand side, Terms without Why would just love with a squared sine squared data We can then So we can simplify that little bit more because this is just why Square one minus sign squares, data and so one minus science court date is co sign squared data. And so we get this expression, and then we can solve that. You can solve that for why square and we find out that why square it is exactly the expression that we were given. And I'll let you do the same thing for X squared. It's just a matter of substituting this in there here, rearranging some things and solving for X squared. And we'll find that that we get a very similar expression for, um for why, over X

All right, So this oneness assets what data means, um for our iconic section graph Arco tension of two data equals a minus c over B uh, fate Ah is just gonna be how much our figure is rotated eso It's our rotation of Arconic section, I think, in a counterclockwise way. Thinking of the figure. Third time's sanction. Okay, that's it. Thank you, very

The definition of the duck product is well, a thought product would be Remember, these are vectors. So now they are going to be cool. The magnitudes multiplied by with each other, and that will be multiplied by the co sign of the angle made in between them. Now to leave close ember itself year after divide both sides by yes, the magnitude of both of these vectors once played by each other. So that would cross out the top and bottom of the right hand side. Leaving us with coast annotate is equal to well, the vector a dot product with the vector be divided by the magnitude of each of defectors. Don't play by each other. So this is our general formula for Nikos Anna Data now using our general definition, we confined when data is equal. Zero R Well, when anything happens so looking at looking at our definition, if the dot product or the top of the left hand side is equal to zero, then there bottom would be a core zero as well. Since you're a developer, anything as he could a zero reveals that just zero is equal to co Santa data soco Santa date is actually go to the X value of the units or Corey. So when is X value could a zero? Well, it is one here, and Brad, two of two over here. But it is. You go to zero at this point right here, And that is equal to pi over two. Also known as many degrees visual eggs. Visualizing this, you will have two vectors and 90 degrees. If they're, that part is equal to zero. Meaning there perpendicular. They're perpendicular winded up. Product is equal to zero.

In this question, it was given that theater equal 30 degree. So extension equal excuse in theater Plus Why science era and why Dash Taken negative X science theater Plus Why Cousin Theater and X equals exquisite. If this goes on Peter minus, Why did science Feta? Why x yes, Science Theater plus y cousin theater. For that, we can see X Dash will equal to square root off three times X plus y over to and why that should equal to negative X plus y squared with three over to an X equals square root of three. Extension in a Swedish over to and why equal extends. Plus why dashes square root of three over to in 0.8 in X Y coordinates system. It was given 4.1 and negative square out of three. So X stage for equal to square root of three Yes, negative schools or three over two. It will equal to zero and why Dashwood equal to negative one. Just square you 23 times negative square root of street over to or equal to negative one minus tree over to for equal to negative two. So the point extension one dish for the equal to zero and negative, too, So point B two X square plus two squared or two or three X y equals three. Why substitute two times square toe three X dash minus y dish over to square, plus to school to three times square root of three. Extension. Find a Swedish over to times X dash plus square root of three y. Dash over to equal to three for that two more times with three extends square. That's why there's a square minus two. Squared 2 to 3 X The short dish over four most to score it'll 3/4. School to three. Stash square minus square to three. Why Dashes Square? Yes to extend. Why? Dash equals to shrink. We get three X dishes square minus. Why they should square equal free. So to get the equation, it will be extension square over one minus. Why, that's just square over three equal one. It presents. So it's we draw the coordinates. Why an ex? And that coordinates off extension. Why Dash? Why dish mixed ish? Well, did you here to be doing? He drew the talking. Give us that result and thank you


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