 Hi, and how are you all today? The question says prove that the curve x is equal to y square and xy is equal to k cut at y-tangles if 8k square is equal to 1. Now here we are given two equations that is x is equal to y square and xy is equal to k. Now here let this be the first equation and this be the second equation. Now putting the value 1 in second, we get that is if we substitute the value of x as y square in the second equation we have y square into y is equal to k that implies y cube is equal to k that gives us the value of y as k raised to the power 1 by 3 and if the value of y is equal to k raised to the power 1 by 3 then the value of x is equal to y square that will be k raised to the power 1 by 3 into 2 that is further equal to x is equal to k 2 by 3. So now here we can say that therefore intersection k raised to the power 2 by 3 that is x comma k raised to the power 1 by 3 which is seen as x comma y and we have assumed it. Now from first we get x is equal to y square. Now on differentiating this equation with respect to x we have 1 is equal to 2y dy by dx or we can see that dy by dx is equal to 1 upon 2 y at this be the third equation. Now also let this be m1. Now from second equation we get x y is equal to k. Now on differentiating it with respect to x we have x that is on using product truth we have first function to derivative of second plus second function to derivative of first that is 1. Zero because derivative of a constant is equal to zero. So we can write it as therefore dy by dx is equal to minus y by x and let this be m since the curves that is 1 first and second cut each other at right angles at their intersection that we have above as x y let's say so we say that therefore m1 into m2 should be equal to minus 1. Let us substitute the value of m1 and m2 from third and fourth equation. So m1 is found out above as 1 upon 2y into m2 as minus y upon x that is equal to minus 1. Or we have minus 1 upon 2x equal to minus 1 that further implies 2x is equal to 1 that is 2 into we know that value of x that we have found out above in respect to k was k raised to the power 2 by 3 right it is equal to 1 that further implies if we cube both the sides we have 2k 2 by 3 the whole cube equal to 1 cube that is after cubing both sides. So now we have 8k square equal to 1 and this is what we were required to prove. So we can write that we hence we have proved the given question. So this completes the session hope you understood it well and enjoyed it too. Do you remember that if two curves cut each other at right angles at their intersection x y? So m1 into m2 that is where both curves slope is equal to minus 1 right bye for now.