For an integer n ≥ 2, if the arithmetic mean of all coefficients in the binomial expansion of (x+y)²ⁿ⁻³ is 16 , then the distance of the point P(2 n-1, n²-4 n) from the line x+y=8 is

Solution

Maths sir · 5 steps

Mean of the coefficients is sum over count: that gives n, then the distance formula does the rest.

1. Sum and count. Put x = y = 1: the coefficients of (x+y)²ⁿ⁻³ add up to 2²ⁿ⁻³. An expansion of power m has m+1 terms, so there are 2n-2 coefficients.

   2²ⁿ⁻³/(2n-2) = 16

2. Find n. Try values: n = 4 gives 32/6, and n = 5 gives 2⁷/8 = 128/8 = 16. The left side keeps growing after this, so n = 5 is the only solution.
3. Locate P. With n = 5, P = (2 × 5 - 1, 25 - 20) = (9, 5).
4. Distance. For the line x + y - 8 = 0, the distance is (|9 + 5 - 8|)/√(1² + 1²) = 6/√2 = 3√2.
5. So. The distance of P from the line is 3√2, which is option (d).

The idea to keep. Sum of the coefficients of (x+y)ᵐ is 2ᵐ and there are m+1 of them. When the unknown sits in both an exponent and a linear term, test small integers.

Answer from the NTA final answer key. Solution by PrepWiser.