If the point on the curve y² = 6x, nearest to the point ( 3,3/2 ) is (α, β), then 2(α + β) is equal to _____________.

Solution

Maths sir · 4 steps

Write the distance using y alone, since x = y²/6, and minimise it. The cubic collapses to y³ = 27.

1. One variable. Any point on the curve is (y²/6, y). Minimise the square of the distance to (3, 3/2) instead of the distance itself. It has the same minimum point and no square root.

   D(y) = (y²/6 - 3)² + (y - 3/2)²

2. Differentiate. D'(y) = 2(y²/6 - 3)y/3 + 2(y - 3/2) = y³/9 - 2y + 2y - 3 = y³/9 - 3. Setting D'(y) = 0 gives y³ = 27, so y = 3.
3. Check minimum. D'(y) < 0 for y < 3 and D'(y) > 0 for y > 3, so y = 3 is the minimum. Then x = 9/6 = 3/2, so (α, β) = (3/2, 3).
4. So. 2(α + β) = 2(3/2 + 3) = 2 × 9/2 = 9.

The common slip: 4.5. Stopping at α + β = 9/2 gives 4.5. The question asks for twice that sum.

The idea to keep. For the nearest point on a curve, write the squared distance in one variable and set its derivative to zero. For y² = 4ax, use y as the variable, because x = y²/4a substitutes in neatly.

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