 Hello friends and how are you all doing today? My name is Priyanka. The question says prove that the curves x is equal to y square and xy is equal to k. Cut at right angles if each k square is equal to 1. Now before proceeding on with the solution we should be well versed that two curves intersect at right angles if the tangent point of intersection pendicular each other. The knowledge of this fact is the key idea for this question. Now we are given that the equation of the curve is x is equal to y square. Right, xy is equal to k. Now on differentiating with respect to x we get 1 is equal to 2y dy by dx and here we have let us first solve this one. This means dy by dx is equal to 1 upon 2y and here we have x dy by dx plus y is equal to 0. This gives us dy by dx is equal to minus y by x. Now here we can substitute the value of y as here xy is equal to k so that means y will be equal to k by x. Right, so here we have dy by dx equal to 1 upon 2k by x or it can be written as dy by dx is equal to x upon 2k and here also we have dy by dx equal to minus k upon x was already here. One more x it will be x square. We can write it minus k upon x square. Now we have these slopes for both the curves. Now after studying our key idea we know that the point of intersection we perpendicular to each other and since the curves cut at right angles that means product of the slopes should be equal to what exactly it should be equal to minus 1. Now we have the slopes as x upon 2k getting multiplied by minus k upon x square should be equal to minus 1. We have 1 upon 2x equal to 1 or we can see that 2x is equal to 1 right or we can say that x is equal to 1 upon 2. Now when x is equal to 1 upon 2 we have the value of y as or before doing this step we were given that x is equal to y square and xy is equal to k. On squaring both the sides we have x square equal to y raised to the power 4 x square y square is equal to k square. Now we can see that the value of y square is equal to x so we can write it as x square into x equal to k square. This is because y square is equal to x which is x cube is equal to k square right. Now we have proved above that 2x is equal to 1 right. Now on cubing both the sides it will be 8x cube equal to 1 right. Now putting the value of x cube as k square we have 8k square equal to 1 right and this is what we were required to prove in this question. So we can write that hence we have proved that 8k square is equal to 1 right. This completes the session hope you understood it well and enjoyed it have a nice day.