Prove that the curves xy = 4 and x2+y2 = 8 touch each other.


Given: two curves x2 + y2 = 8 and xy = 4


To prove: the two curves touch each other


Explanation:


Now given x2 + y2 = 8


Differentiating this with respect to x, we get



Applying sum rule of differentiation, we get






Also given xy = 4


Differentiating this with respect to x, we get



Now applying the product rule of differentiation, we get






But the two curves touch each other, if


m1 = m2


Now substituting the values from equation (ii) and equation (ii), we get



y2 = x2


x = y…….(iii)


Now substituting x = y in x2 + y2 = 8, we get


y2 + y2 = 8


2 y2 = 8


y2 = 4


y = ±2


When y = 2,


xy = 4 becomes


x(2) = 4 x = 2


when y = -2,


xy = 4 becomes


x(-2) = 4 x = -2


Hence the point of intersection of the two curve is (2,2) and (-2, -2)


Substituting these points of intersection equation (i) and equation (ii), we get


For (2,2),




m1 = m2


For (-2,-2),




m1 = m2


Therefore, for both curves to touch each other, the slopes of both the curves should be same.


Hence the two given curves touch each other.


Hence proved


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