Question

Solve it fast plz...


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Answer 1

Step-by-step explanation:

As per the given questions

Let P (x₁ = −3, y₁ = 9), Q(x₂ = a, y₂ = b) and R(x₃ = 4, y₃ = −5) be the given points.

The given points are collinear if

x₁ (y₂ − y₃) + x₂ (y₃ − y₁) + x₃ (y₁ − y₂) = 0

Now put the given values, we get

-3 (b + 5) + a (−5 − 9) + 4 (9 − b) = 0

⇒ −3b − 15 − 14a + 36 − 4b = 0

⇒ 2a + b = 3 ......................equation (i)

And a + b = 1 (given) ..........equation (ii)

Now, on subtracting (i) - (ii), we get

a = 2

Substitute this value in equation (ii), we get

2 + b = 1

b = -1

Hence, a = 2 and b = −1.

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Answer 2

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✤ Required Answer:

✒ GiveN:

• Points are P(-3, 9), Q(a,b) and R(4,-5)
• These points are collinear
• a + b = 1

✒ To FinD:

• Find the value of a and b....?

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✤ How to Solve?

Basically, Collinear points refer to the points that lie on the same straight line. Sum of distance between any two points of two pairs is equal to length of distance between the third pair.

• So, we can say that it have only one dimension and hence, the area is 0 because no triangle or any polygon is formed.

By using formula for area,

• Let given points (x1, y1) , (x2, y2) and (x3, y3) are collinear, then the area of triangle formed by them is 0.

➤ Area of triangle = 0

${ \sf{ \rightarrow \frac{1}{2} \bigg (x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \bigg ) = 0}}$

$\sf{ \rightarrow \: x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) = 0}$

✒ So, By using this condition for three collinear points, let's solve this question....

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✤ Solution:

Let,

• x1 = -3 and y1 = 9
• x2 = a and y2 = b
• x3 = 4 and y3 = -5

By applying the above condition,

$\sf{ \rightarrow x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) = 0}$

$\sf{ \rightarrow - 3(b - ( - 5)) + a( - 5 - 9) + 4(9 - b) = 0}$

$\sf{ \rightarrow - 3(b + 5) + a( -14) + 4(9 - b) = 0}$

$\sf{ \rightarrow - 3b - 15 - 14a + 36 - 4b = 0}$

$\sf{ \rightarrow 21 - 14a - 7b = 0}$

$\sf{ \rightarrow 21 = 14a + 7b}$

$\sf{ \rightarrow 7(2a + b) = 21}$

$\sf{ \rightarrow 2a + b = 3 .........(1)}$

Given in the question,

• a + b = 1..............(2)

Subtracting eq(2) from eq(1),

$\sf{ \rightarrow 2a + b - (a + b) = 3 - 1}$

$\sf{ \rightarrow 2a + b - a - b = 2}$

$\sf{ \rightarrow a = 2}$

Putting value of a in eq(2),

$\sf{ \rightarrow 2 + b = 1}$

$\sf{ \rightarrow b = - 1}$

❇ Hence, Required values of a and b

• a = 2
• b = -1

✒ Hence, solved !!

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