Q21 of 157 Page 753

Prove that the points A(a, 0), B(0, b) and C(1, 1) are collinear, if

To show that the points are collinear, we show that the area of triangle is equilateral = 0

Δ = 0

Δ = 1/2{x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)} = 0

⇒ Δ = 1/2{a(b– 1) + 0 (1– 0) + 1 (0– b)} = 0

⇒ (ab–a–b) = 0

Dividing the equation by ab.

1–1/b–1/a

1–(1/a + 1/b)

1–1 = 0

Hence collinear.

Find a relation between x and y, if the points A(2, 1), B(x, y) and C(7, 5) are collinear.

Find a relation between x and y, if the points A(x, y), B(–5, 7) and C(–4, 5) are collinear.

If the points P(–3, 9), Q(a, b) and R(4, –5) are collinear and a + b = 1, find the values of a and b.

Points A(-1, y) and B(5, 7) lie on a circle with centre O(2, - 3y). Find the values of y.