Find a point on the curve xy + 4 = 0 at which the tangents are inclined at an angle of 45^o 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)