Question

If the lines 3x + by + 5 = 0 and ax – 5y + 7 = 0 are perpendicular to each other, find the relation between a and b.

Answer 1

For two lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 to be perpendicular, the sum of the products of their x and y coefficients must be zero:

(a₁ × a₂) + (b₁ × b₂) = 0

Coefficients from the equations:

• Line 1 (3x + by + 5 = 0): a₁ = 3, b₁ = b
• Line 2 (ax – 5y + 7 = 0): a₂ = a, b₂ = −5

Plug into the formula:

(3 × a) + (b × −5) = 0

3a − 5b = 0 or 3a = 5b