If the points (a,b),(3,−5) and (−5,−2) are collinear. Then find the value of 3a+8b

Question

Filo · Accepted Answer

If three points are collinear then the area of the triangle is zero.

Area of the triangle passing through the points

(x_{1}, y_{1})

,

(x_{2}, y_{2})

and

(x_{3}, y_{3})

is given by:

A = \frac{1}{2} ∣ x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) ∣

Here the points are

(a, b)

,

(3, - 5)

and

(- 5, - 2)

and it is given that these points are collinear therefore the area is zero that is:

A = \frac{1}{2} ∣ a (- 5 - (- 2)) + 3 (- 2 - b) + (- 5) (b - (- 5)) ∣ \Rightarrow 0 = \frac{1}{2} ∣ a (- 5 + 2) + 3 (- 2 - b) - 5 (b + 5) ∣ \Rightarrow 0 = \frac{1}{2} ∣ - 3 a - 6 - 3 b - 5 b - 25 ∣ \Rightarrow 0 = \frac{1}{2} ∣ - 3 a - 8 b - 31 ∣ \Rightarrow 0 = - 3 a - 8 b - 31 \Rightarrow 3 a + 8 b = - 31

Hence,

3 a + 8 b = - 31

.

Question Text	If the points $(a, b), (3, - 5)$ and $(- 5, - 2)$ are collinear. Then find the value of $3 a + 8 b$
Answer Type	Text solution:1
Upvotes	150

If the points $(a, b), (3, - 5)$ and $(- 5, - 2)$ are collinear. Then find the value of $3 a + 8 b$