Question

Find the condition, when

(i) Line y = mx + c, touches the circle (x – a)² + (y – b)² = r²

(ii) Line lx + my + n = 0 touches the circle x² + y² = a²

Answer 1

(i) Equation of line

y = mx + c …(i)

Equation of circle (x – a)² + (y – b)² = r²

Centre of circle = (a, b) and Radius = r

Line will touch (i) and (ii), if

Radius of circle = length of ± drawn from centre to line

⇒ (1 + m²)r² = m²a² + b² + c² – 2mab – 2bc + 2mac

⇒ r² + m²r² = m²a² + b² + c² – 2mab – 2bc + 2mac

⇒ m² – a² – m²r² + 2mac – 2mab + b² + c² – 2bc = r²

⇒ (a² – r²)m² + (2ac – 2ab)m + b² + c² – 2bc = r²

⇒ (a² – r²)m² + 2a(c – b)m + (b – c)² = r²

⇒ m²(a² – r²) + 2ma(c – b) + (c – b)² = r²

(ii) Equation of given line

lx + my + n = 0

⇒ my = – lx – n