Question

if the points p(-3,9),q(a,b)and r(4,-5) are collinear and a + b = 1,find the values of a and b.

Answer 1

If Point P, Point Q and Point R are collinear points then there slope will be equal.

Also,
Slope = (y₂ - y₁)/(x₂ - x₁)

In PQ,

Let y₂ = b, y₁ = 9, x₂ = a, x₁ = -3

Slope of PQ =
$\frac{b - 9}{a - ( - 3)} = \frac{b - 9}{a + 3}$

For RQ,

Let y₂ = b, y₁ = -5, x₂ = a, x₁ = 4

Slope of RQ =
$\frac{b - ( - 5)}{a - 4} = \frac{b + 5}{a - 4}$
Now as they are collinear,

Slope of PQ = Slope of RQ
$\frac{b - 9}{a + 3} = \frac{b + 5}{a - 4}$
∴ (b-9)(a-4)=(b+5)(a+3)
∴ ab - 9a - 4b + 36 = ab + 5a + 3b + 15
∴ 36 - 15 = 5a + 9a + 3b + 4b
∴ 21 = 14a + 7b
∴ 3 = 2a + b
∴ b = 3 - 2a ----------------------(Equation 1)

Now, We had given that

a + b = 1

Putting Equation 1 in this

a + 3 - 2a = 1
-a = -2
∴ a = 2 -------------------(Result 1)

Now putting value of "a" in Equation 1

b = 3 - 2a
b = 3 - 2(2)
b = 3 - 4
b = -1

Answer:- a = 2 and b = -1