If (0, 0), (4, –3) and (x, y) are collinear then

Let A = (0, 0), B = (4, -3) and C = (x, y)

As the points are collinear area of triangle formed by these points is 0

Area = × [x₁(y₂ – y₃) + x₂(y₃ – y₁) + x₃(y₁ – y₂)]

Where (x₁, y₁), (x₂, y₂) and (x₃, y₃) are vertices of triangle

Here,

(x₁, y₁) = (0, 0)

(x₂, y₂) = (4, -3)

(x₃, y₃) = (x, y)

Substituting values

⇒ 0 = × [0(-3 – y) + 4(y – 0) + x(0 – (-3))]

⇒ 0 = 0 + 4y + 3x

⇒ 4y + 3x = 0

Here there are infinite values for x and y which will satisfy the equation 4y + 3x = 0 which means there are infinite points

This can also be seen geometrically that is if line is passing through (0, 0) and (4, -3) there are infinite points on this line.

We can select the correct option by substituting the values given in option in equation 4y + 3x = 0 and if it satisfies then that is the correct option

x = 8, y = -6 satisfies the equation 4y + 3x = 0

x = –8, y = 6 satisfies the equation 4y + 3x = 0

Now, C = (8, -6) or C = (-8, 6), but for A, B and C to be collinear

AB + BC = AC
We know, By distance formula, Distance between two points X(x₁, y₁) and Y(x₂, y₂) is

Case I: C = (8, -6)

AB

units

BC

units

AC

units

Clearly, AB + BC = AC

Hence, (0, 0), (4, -3) and (8, -6) are collinear

Case II: C = (-8, 6)

AB

units

BC

units

AC

units

Clearly, AB + BC ≠ AC

Hence, (0, 0), (4, -3) and (8, -6) are not collinear.

[In this case, BA + AC = BC, ⇒ B, A and C are collinear]

Hence, Correct option is (a)