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Given: points (k, 3), (6, - 2) and ( - 3, 4) are collinear

To find: The value of k.

Formula Used:

Area of the triangle having vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃)
= 1/2 |x₁(y₂ - y₃) + x₂(y₃ – y₁) + x₃(y₁ – y₂) |

Explanation:

The three given points A (k, 3), B (6, −2) and C (−3, 4) are collinear.

Collinear means points are in straight line which implies that they cannot make a triangle.

Hence, the area of the triangle formed by collinear points will be zero.

For the given points,

∴ 0 = 1/2 k ( - 2 – 4) + 6(4 - 3) - 3(3 – ( - 2)) |

∴ 0 = 1/2 | - 6k + 6 - 15|

∴ 0 = 1/2 | - 6k - 9|

∴ - 1/2 |6k + 9| = 0

6k + 9 = 0

∴ k =

Hence, the value of k is .

Note: You can also prove collinearity using distance formula, but the method for the area of the triangle makes the solution simpler.

You can attempt these questions with an area of triangle formula unless you are asked to solve by other methods.