If the tangent to the curve y=x³-x²+x at the point (a, b) is also tangent to the curve y = 5x² + 2x - 25 at the point (2, -1), then |2a + 9b| is equal to __________.

Solution

Maths sir · 4 steps

Build the tangent from the parabola first, then hunt for where the cubic has that same slope and lies on it.

1. Tangent at (2, −1). For y = 5x² + 2x - 25, dy/dx = 10x + 2, which is 22 at x = 2. The tangent line is y + 1 = 22(x - 2), that is, y = 22x - 45.
2. Match the slope. For the cubic y = x³ - x² + x, the slope at x = a is 3a² - 2a + 1. Set it equal to 22.

   3a² - 2a - 21 = 0 ⇒ (a - 3)(3a + 7) = 0 ⇒ a = 3 or a = -7/3

3. Check on the line. At a = 3, b = 27 - 9 + 3 = 21, and the line gives 22 × 3 - 45 = 21, so it fits. At a = -7/3, the cubic gives b = -553/27 but the line gives -289/3 = -2601/27, so that point is not on the tangent.
4. So. (a, b) = (3, 21) and |2a + 9b| = |6 + 189| = 195.

The common slip: 189. Keeping the root a = -7/3 without checking that it lies on the line gives b = -553/27 and |2a + 9b| = 189, not 195.

The idea to keep. A common tangent needs two things at the point: the same slope and the point lying on the line. Equal slope alone can give extra roots, so always check each one against the line.

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