Question

The equation of the path traversed by the ball headed by the footballer is y = ax² + bx + c (where 0 ≤ x ≤ 14 and  and a, b, c ∈ R and a ≠ 0) with respect to a XY-coordinate system in the vertical plane. The ball passes through the points (2, 15), (4, 25) and (14, 15). Determine the values of a, b and c by solving the system of linear equations in a, b and c using the matrix method. Also find the equation of the path traversed by the ball.

Answer 1

y = ax² + bx + c

15 = 4a + 2b + c

25 = 16a + 4b + c

15 = 196a + 14b + c

The set of equations can be represented in the matrix form as AX = B,

where `A = [(4, 2, 1), (16, 4, 1), (196, 14, 1)], X = [(a), (b), (c)] and B = [(15), (25), (15)]`

⇒ `[(4, 2, 1),(16, 4, 1), (196, 14, 1)][(a), (b), (c)] = [(15), (25), (15)]`

|A| = 4(4 − 14) −2 (16 − 196) + (224 − 784)

= − 40 + 360 − 560

= −240 ≠ 0

Hence A⁻¹ exists.

Now, adj(A) = `[(-10, 180, -560), (12, -192, 336), (-2, 12, -16)]^T` = `[(-10, 12, -2), (180, -192, 12), (-560, 336, -16)]`

 `[(a), (b), (c)] = -1/240[(-10, 12, -2), (180, -192, 12), (-560, 336, -16)][(15), (25), (15)]`

= `-5/240[(-10, 12, -2), (180, -192, 12), (-560, 336, -16)][(3), (5), (3)]`

= `-5/240[(24), (-384), (-48)]`

`therefore a = -1/2, b = 8, c = 1`

So, the equation becomes `y = -1/2x^2 + 8x + 1`