Question

If (4,a),(6,11) and (a,13) are collinear,then find the value of a?

Answer 1

Question →

→If (4,a),(6,11) and (a,13) are collinear,then find the value of a?

Answer:-

$\implies \: \boxed{a \: = 10 \: or \: 7}$

Step - by - step explanation→

Used property→

Here we used property of collinear points .

$let \: \: points \: (x_1, \: y_1) \: (x_2 \: ,y_2) \: (x_3 ,\: y_3) are \: \\ collinear \: .then \\ 0 = \frac{1}{2} \bigg(x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \bigg)$

Solution :-

Given points are (4,a) ,(6,11) ,(a,13).

Let ,

$x_1 = 4 ,\: y_1 = a \\ \\ x_2 \: = 6 ,\: y_2 = 11 \\ \\ x_3 = a, \: y_3 = 13$

According to the question,

Points are collinear then,

$0 = \frac{1}{2} \bigg( 4(11 - 13) + 6(13 - a) + a(a - 11) \bigg) \\ \\ 0 = - 8 + 78 - 6a + {a}^{2} - 11a \\ \\ 0 = {a}^{2} - 17a + 70 \\ \\ 0 = {a}^{2} - (10 + 7)a + 70 \\ \\ 0 = {a}^{2} - 10a - 7a + 70 \\ \\ 0 = a(a - 10) - 7(a - 10) \\ \\ 0 = (a - 10)(a - 7) \\ \\ \star \: case \: (1) \\ if \: \: \\ \implies \: (a - 10) = 0 \\ \\ \implies \: \boxed{a = 10 }\\ \\ \star \: case \: (2) \\ if \\ \implies \: (a - 7) = 0 \\ \\ \implies \: \boxed{ a = 7}$

Hence,

$the \: value \: of \: a \: is \: 10 \: or \: 7.$