Question

10. By using the concept of the equation of the straight line, prove that the given three
points are collinear.
(i) (4, 2), (7, 5) and (9, 7) (ii) (1, 4), (3, -2) and (-3, 16)

Answer 1

if three points A, B and C are collinear then, slope of line joining points A and B = slope of line joining the points B and C = slope of line joining the points C and A .

(i) (4,2), (7,5) and (9,7)
Let A = (4,2) , B = (7,5) and C = (9,7)
slope of line joining A and B = (5 -2)/(7 - 4)=1
slope of line joining B and C = (7- 5)/(9 -7)=1
slope of line joining C and A = (7 -2)/(9 -4) = 1
here it is clear that, slope of line joining A and B = slope of the joining B and C = slope of line joining C and A = 1
so, points A , B and C are collinear.

(ii) Let A = (1,4) , B = (3, -2) and C= (-3,16)
slope of line joining A and B = (-2-4)/(3-1)=-3
slope of line joining B and C = (16+2)/(-3-3)=-3
slope of line joining C and A = (16-4)/(-3-1)=-3
here it is clear that, slope of line joining A and B = slope of the joining B and C = slope of line joining C and A = -3
so, points A, B and C are collinear.

Answer 2

Solution :

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Equation of the line passing

through ( x1, y1) and (x2 , y2 ) is

(y - y1)= [( y2 - y1)/(x2-x1) ](x-x1)

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i ) Given A(4,2)=(x1,y1),

B(7,5)=(x2 ,y2 ) ,

C(9,7)

Equation of a line AB

y-2 = [(5-2)/(7-4)](x-4)

=> y-2 = (3/3)(x-4)

=> y-2 = x-4

=> -2 + 4 = x - y

=> x - y = 2-----( 1 )

Substitute C( 9 , 7 ) in equation ( 1 ),

9 - 7 = 2

=> 2 = 2 ( True )

Therefore ,

A, B and C are collinear .

ii ) Given A(1,4 ) , B(3,-2), C( -3,16 )

Equation of a line AB ,

y - 4 = [(-2-4)/(3-1)](x-1)

=> y - 4 = (-6/2)(x-1)

=> y-4 = -3(x-1)

=> y - 4 + 3(x - 1 ) = 0

=> y - 4 + 3x - 3 = 0

=> 3x + y - 7 = 0---( 2 )

Now , substitute C(-3,16) in equation

( 2 ) ,

3(-3) + ( 16 ) - 7 = 0

=>-9 + 16 - 7 = 0

=> -16 + 16 = 0

0 = 0 ( True )

Therefore ,

A , B and C are collinear points .

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