Question

A boy is walking along the path y = ax² + bx + c through the points (– 6, 8), (– 2, – 12), and (3, 8). He wants to meet his friend at P(7, 60). Will he meet his friend? (Use Gaussian elimination method.)

Answer 1

We are given y = ax² + bx + c

Also we are given (-6, 8), (-2, -12) and (3, 8) are points on the path.

(i) (– 6, 8) is a point on y = ax² + bx + c

At x = – 6, y = 8

(i.e) a(36) + b(– 6) + c = 8

⇒ 36a – 6b + c = 8   .......(1)

(ii) (– 2, – 12) is a point on y = ax² + bx + c

At x = – 2, y = – 12

⇒ a(– 2)² + b(– 2) + c = – 12

⇒ 4a – 2b + c = – 12   ......(2)

(iii) (3, 8) is a point on y = ax² + bx + c

At x = 3, y = 8

⇒ a(3)² + 6(3) + c = 8

⇒ 9a + 3b + c = 8   ......(3)

The matrix form of the above three equations is

`[(36, -6, 1),(4, -2, 1),(9, 3, 1)][("a"),("b"),("c")] = [(18),(-12),(8)]`

(i.e) AX = B

The augmented matrix (A, B) is

[A, B] = `[(36, -6, 1, 8),(4, -2, 1, -12),(9, 3, 1, 8)]`

`˜ [(36, -6, 1, 8),(0, -12, 8, -116),(0, 18, 3, 24)] {:("R"_2 -> 9"R"_2 - "R"_1),("R"_3 -> 4"R"_3 - "R"_1):}`

`˜ [(36, -6, 1, 8),(0, -12, 8, -116),(0, 0, 30, -300)] "R"_3 -> 2"R"_3 + 3"R"_2`

The above matrix is in echelon form.

Now writing the equivalent equations we get

`[(36, -6, 1),(0, -12, 8),(0, 0, 30)][("a"),("b"),("c")] = [(8),(-116),(-300)]`

(i.e) 36a – 6b + c = 8

⇒ – 12b + 8c = – 116

⇒ 30c = – 300

⇒ c = – 10

Substituting c = -10 in (2) we get

– 12b + 8(– 10) = – 116

⇒ – 12b = – 116 + 80 = – 36

⇒ b = 3

Substituting c = – 10, b = 3 in (1) we get

36a – 6(3) + (– 10) = 8

⇒ 36a – 18 – 10 = 8

⇒ 36a = 8 + 18 + 10 = 36

⇒ a = 1

a = 1, b = 3 and c = – 10

y = (1)x² + (3)x + (– 10)

y = x² + 3x – 10

Now at x = 7, y = (7)² + 3(7) – 10

= 49 + 21 – 10

= 60

(7, 60) is a point on the path so he will meet his friend.