Find the condition that the curves and 2xy = k intersect orthogonally.
Given: two curves and 2xy = k
To find: to track the condition where both the curves intersect orthogonally
Explanation: Given 2xy = k
Put in the value of y in another curve equation, i.e., we get
Putting both the sides under cube root, we get
Substituting equation (ii) in equation (i), we get
is the point of intersection of two curves
Now given
After differentiating the equation for x, we get
After tracking the value of differentiation at the point of intersection, i.e., at we
get
Also given 2xy = k
Differentiating this with respect to x, we get
Then, again, finding the given differentiation value at the point of intersection, i.e., at , we get
However, the orthogonal interaction of two curves occurs if
m1.m2 = -1
Then, Substituting the values from equation (iii) and (iv), we get
This condition proves to fulfil the orthogonal interaction point for the two curves.