Question

Suppose the circles x² + y² = 1 and (x - 1)² + (y - 1)² = r² intersect each other orthogonally at the point (u, v). Then u + v = ______.

Answer 1

Concept:

Condition for orthogonal:

m1 ⋅ m2 = - 1

Where, m1 = Slope of curve-1 

m2 = Slope of curve-2

Calculation:

Given:

Curve-1: 

x2 + y2 = 1

Differentiating both side,

\(2x\;+\;2y{dy\over{dx}} =0 \)

\({dy\over dx}=m_1={-x\over y}\)

Curve-2: 

(x - 1)2 + (y - 1)2 = r2

Differentiating both side,

\(2(x-1)+2(y-1){dy\over dx} = 0\)

\({dy\over dx}=m_2 ={{1-x}\over {y-1}}\)

Both curves are intersecting each other at the point (u,v), so x and y are replaced by u and v.

m1 ⋅ m2 = - 1

\(m_1 \cdot m_2 = {[{{({-u\over v})}{({1-u})\over {({v-1})}}}]}=-1\)

- u + u2 = - v2 + v

u2 + v2 = u + v

∵ x2 + y2 = 1 ⇒ u2 + v2 = 1

∴ u + v = 1