Question

Angle of intersection of two polar curves is equal to the angle between the tangents drawn at the point of intersection of the two curves then What is the condition for the two curves intersecting orthogonally for the below mentioned figure?

(a) tan φ1⁡ . tan⁡φ3=1

(b) tan⁡φ1 . tan⁡φ3=-1

(c) tan⁡(φ1+φ3)=\(\frac{1}{\sqrt{2}}\)

(d) tan⁡(φ1-φ3)=\(\frac{1}{\sqrt{2}}\)

I have been asked this question by my school teacher while I was bunking the class.

My question comes from Polar Curves topic in chapter Differential Calculus of Engineering Mathematics

Answer 1

Right answer is (b) tan⁡φ1 . tan⁡φ3=-1

Easy explanation: From the figure & according to orthogonal principle |φ1-φ3|=\(\frac{π}{2}\)

further φ1=\(\frac{π}{2}\)+φ3……taking tangent of the angle

\(tan⁡ φ1 = tan(⁡\frac{π}{2}+φ3) = -cot⁡φ3 = -\frac{1}{tan⁡φ3} \)

tan⁡φ1 .tan⁡φ3 = -1.