Question

if the points A(-2,1) B(a,b) C(4,-8) are collinear and a-b =1 find the value of a and b​

Answer 1

Answer:

As one equation in the form of a and b is given in the question and there are two variable so we need at least two equation to find the value of a and b.

Second equation comes by putting the three points are collinear. Then after solving both of them simultaneously we will get the value of a and b

As:-

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If A(-2,1) B(a,b) and C(4,-8)

are collinear then area of triangle formed by joining A , B and C that is ar. △ABC = 0.

So,

$\frac{1}{2} | - 2(b + 8) + a( - 8 - 1) + 4(1 - b)| = 0 \\ \implies | - 2b - 16 - 8a - 8 + 4 - 4b| = 0 \\ \implies | - 8a - 6b - 20| = 0 \\ \implies - 8a - 6b - 20 = 0 \\ \implies8a + 6b + 20 = 0 \\ \implies4a + 3b = - 10 \: \: \: \: \: \: \: ..........(i)$

Also,

Given that,

a - b = 1 ......... (ii) .

Now,

Solving (i) and (ii) simultaneously

we get,

a = -1

and

b= -2.

Answer 2

Answer:

If A(-2,1) B(a,b) and C(4,-8)

are collinear then area of triangle formed by joining A , B and C that is ar. △ABC = 0.

So,

$\frac{1}{2} | - 2(b + 8) + a( - 8 - 1) + 4(1 - b)| = 0 \\ \implies | - 2b - 16 - 8a - 8 + 4 - 4b| = 0 \\ \implies | - 8a - 6b - 20| = 0 \\ \implies - 8a - 6b - 20 = 0 \\ \implies8a + 6b + 20 = 0 \\ \implies4a + 3b = - 10 \: \: \: \: \: \: \: ..........(i)$

Also,

Given that,

a - b = 1 ......... (ii) .

Now,

Solving (i) and (ii) simultaneously

we get,

a = -1

and

b= -2.