Question

if the points p (-3,9) Q(a,B) and R(4,-5) are collinear and a+b=1, then find the value of a and b.​

Answer 1

Step-by-step explanation:

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}

a−(−3)

b−9

=

a+3

b−9

For RQ,

Let y₂ = b, y₁ = -5, x₂ = a, x₁ = 4

Slope of RQ =

\frac{b - ( - 5)}{a - 4} = \frac{b + 5}{a - 4}

a−4

b−(−5)

=

a−4

b+5

Now as they are collinear,

Slope of PQ = Slope of RQ

\frac{b - 9}{a + 3} = \frac{b + 5}{a - 4}

a+3

b−9

=

a−4

b+5

∴ (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