If 3x − ky + 8 = 0 and 9x − 18y + 20 = 0 have no solution, find k.

Correct — D. Two lines have no solution when they are parallel but distinct: a₁⁄a₂ = b₁⁄b₂ ≠ c₁⁄c₂.

Ratio of the x-coefficients: 3⁄9 = 1⁄3
Ratio of the y-coefficients: (−k)⁄(−18) = k⁄18
Set them equal: k⁄18 = 1⁄3, so k = 18 ÷ 3 = 6

Check the constants: 8⁄20 = 2⁄5, which is not 1⁄3, so the lines never meet → option (d)

Why the others are wrong

• (a)
  9 — With k = 9 the y-ratio is 9⁄18 = 1⁄2, not the 1⁄3 of the x-ratio. The slopes differ, so the lines cross at exactly one point.
• (b)
  10 — k = 10 gives a y-ratio of 10⁄18 = 5⁄9 against the x-ratio 1⁄3. Unequal ratios mean one crossing point, a unique solution, the opposite of what is asked.
• (c)
  2 — k = 2 gives a y-ratio of 2⁄18 = 1⁄9, a third of the x-ratio 1⁄3. The lines are not parallel, so they still meet once.

Each equation ax + by + c = 0 is a straight line, and the ratios of the coefficients decide how two lines meet.

a₁⁄a₂ ≠ b₁⁄b₂: they cross once, a unique solution
a₁⁄a₂ = b₁⁄b₂ = c₁⁄c₂: they are the same line, infinitely many solutions
a₁⁄a₂ = b₁⁄b₂ ≠ c₁⁄c₂: they are parallel, no solution

Key facts

• No solution: a₁⁄a₂ = b₁⁄b₂ ≠ c₁⁄c₂ (parallel, distinct lines).
• Infinitely many solutions: a₁⁄a₂ = b₁⁄b₂ = c₁⁄c₂ (the same line).
• Unique solution: a₁⁄a₂ ≠ b₁⁄b₂ (intersecting lines).

Common traps

• Stopping at a₁⁄a₂ = b₁⁄b₂ without checking the constants: if c₁⁄c₂ matched too, the lines would coincide.
• Inverting one ratio, writing 3⁄9 = 18⁄k, which gives k = 54.

No directly related past PYQ was found.