The angle between the curves C1 and C2 at a point of intersection P is

defined to be the angle between the tangent lines to C1 and C2 at P (if these

tangent lines exist) Let us represent the two curves C1 and C2 by the Cartesian

equation y = f(x) and y = g(x) respectively. Let them intersect at P (x1,y1) .

If ψ1 and ψ2 are the angles made by the tangents PT1 and PT2 to

C1 and C2 at P, with the positive direction of the x – axis, then m1 = tan ψ 1 and

m2 = tan ψ2 are the slopes of PT1 and PT2 respectively.

Let ψ be the angle between PT1

and PT2. Then ψ = ψ2 – ψ1 and

tan ψ = tan (ψ2 – ψ1)

=

tan ψ2 – tan ψ1

1 + tan ψ1 tan ψ2

=

m2 – m1

1 + m1m2

where 0 ≤ ψ < π

We observe that if their slopes are equal namely m1 = m2 then the two

curves touch each other. If the product m1 m2 = – 1 then these curves are said to

cut at right angles or orthogonally. We caution that if they cut at right angles

then m1 m2 need not be –1.

We caution that if they cut at right angles

then m1 m2 need not be –1.

Is this last statement is correct? If yes, please give an example that two curves are cut at right angles and the product of the slopes m1*m2 is not equal to -1.