Find a point on the curve xy + 4 = 0 at which the tangents are inclined at an angle of 45o with the x–axis.

Given:


The curve is xy + 4 = 0


If a tangent line to the curve y = f(x) makes an angle with x – axis in the positive direction, then


= The Slope of the tangent = tan


xy + 4 = 0


Differentiating the above w.r.t x


x(y) + y(x) + (4) = 0


x + y = 0


x = – y


...(1)


Also, = tan45° = 1 ...(2)


From (1) & (2),we get,


= 1


x = – y


Substitute in xy + 4 = 0,we get


x( – x) + 4 = 0


– x2 + 4 = 0


x2 = 4


x = 2


so when x = 2,y = – 2


& when x = – 2,y = 2


Thus, the points are (2, – 2) & ( – 2,2)


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