Question

Determine the value of $a$ such that $(a,a),(2,3)$ and $(4,-1)$ are collinear.

• A. $\frac{3}{4}$
• B. $\frac{5}{3}$
• C. $\frac{1}{3}$
• D. $\frac{7}{3}$

Answer 1

The correct option is D $\frac{7}{3}$

collinear means points in the same line

i.e. if these points are collinear, it means that these points lie on the same line.

now assume that these three points make a triangle..since these points lie on the same line, so the area covered by these three points is zero.

using area of a triangle formula,

area $=1/2\left[x1\right(y2-y3)+x2(y3-y1)+x3(y1-y2\left)\right)]=0$

Therefore Condition for collinearity,

$x_1{y}_2+x_2{y}_3+x_3{y}_1=x_2{y}_1+x_3{y}_2+x_1{y}_3$

Here,

$x_1=a,y_1=a,x_2=2,y_2=3$ and $x_3=4,y_3=-1$
Put this in above equation, we get
$\Rightarrow 3a-2+4a=2a+12-a$
$\Rightarrow 7a-2=12+a\Rightarrow 6a=14\Rightarrow a=14/6$ or $7/3$