Question

The points A(1 , 2), B(3 , 8) and C(x , 20) are collinear if x = *

Answer 1

Given,

The coordinates of the given points are, A(1 , 2), B(3 , 8) and C(x , 20)

To find,

The value of x is the three points are collinear.

Solution,

The area of a straight line formed by three points, is always zero. So, we have to apply the area of triangle formula by the given three points,and if for making them collinear we have to assume that the final answer will be zero.

A = (1,2) = X1,Y1

B = (3,8) = X2,Y2

C = (x,20) = X3,Y3

Area of the triangle :

½ × [X1 (Y2-Y3) + X2 (Y3-Y1) + X3 (Y1-Y2)]

= ½ × [1×(8-20) + 3×(20-2) + x (2-8)]

= ½ × (-12+54-6x)

= ½ × (42-6x)

= (21-3x) square unit

Now if the three points are collinear, then

21-3x = 0

-3x = -21

3x = 21

x = 7

Hence, the value of x will be 7

Answer 2

SOLUTION

GIVEN

The points A(1 , 2), B(3 , 8) and C(x , 20) are collinear

TO DETERMINE

The value of x

EVALUATION

Here the equation of the line passing through the points A(1 , 2), B(3 , 8) is

$\displaystyle \sf{ \frac{y - 8}{x - 3} = \frac{8 - 2}{3 - 1} }$

$\implies \displaystyle \sf{ \frac{y - 8}{x - 3} = \frac{6}{2} }$

$\implies \displaystyle \sf{ \frac{y - 8}{x - 3} = 3}$

$\implies \displaystyle \sf{3x - 9 = y - 8}$

$\implies \displaystyle \sf{3x - y = 1} \: \: \: \: .......(1)$

Since the points A(1 , 2), B(3 , 8) and C(x , 20) are collinear

∴ C(x , 20) is a point on the line joining the points A(1 , 2), B(3 , 8)

∴ C(x , 20) is a point on the line 3x - y = 1

$\therefore \: \sf{3x - 20 = 1}$

$\implies \: \sf{3x = 21}$

$\implies \: \sf{x = 7}$

Hence the required value of x = 7

FINAL ANSWER

If the points A(1 , 2), B(3 , 8) and C(x , 20) are collinear then x = 7

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