Question

If the points (6,0),( K,1) and (- 6,- 1) are collinear then find the value of k.

Answer 1

As the points are collinear the area of triangle will be 0.

Use area of triangle formula of co ordinate geometry and substitute the given values.

Answer 2

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K = 18

=> COLLINEAR POINTS :-

Those points in a plane which lie on the same line are called collinear points.

Let the given collinear pts. be denoted as

A (6,0)

Here,

x1 = 6 and y1 = 0

B (K,1)

Here,

x2 = K and y2 = 1

C (-6,-1)

Here,

x3 = -6 and y3 = -1

Now,

Since the points are collinear,

The triangle formed as ∆ ABC will have area equal to zero.

It means the Determinant will be zero.

=> ∆ = 0

=> 1/2 | x1(y2 - y3) - x2 ( y1- y3) + x3 ( y1 - y2) | = 0

Here, | | sign denotes the modulus so that all the quanties on simplification will be positive, as area can't be negative.

Now, putting the values of x1, x2, x3, y1, y2 and y3, in the Equation,

we get,

=> 1/2 | 6(1+1) - K(0+1) -6 (0-1)| = 0

=> 1/2 | 12 - K + 6 | = 0

Dividing both L.H.S and R.H.S by 1/2,

we get,

=> | 18 - K | = 0

=> 18 - K = 0

=> K = 18