A,b and c are three collinear points such that Ab = bc if the coordinates of a b and c are a,2 1,3 and 5,b find a and b
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Given:
A(a, 2) , B(1,3) & C(5,b) are collinear.
We can say that when the points are collinear, slope are equal.
Slope between AB = Slope between BC = Slope between AC
(2-3)/(a-1) = (3-b)/(1-5) = (2-b)/(a-5)
1/(a-1) = (3-b)/(-4) = (2-b)/(a-5)
1/(a-1) = (b-3)/(4) = (2-b)/(a-5) ——> 1
1/(a-1) = (b-3)/(4)
4 = (a-1)*(b-3)
4 = ab -3a - b + 3
ab -3a - b = 1 ——> 2
(b-3)/(4) = (2-b)/(a-5)
(b-3)* (a-5) = (2-b)*4
ab - 3a -5b + 15 = 8 - 4b
ab - 3a - b = -7 ——> 3
1/(a-1) = (2-b)/(a-5)
(a-5) = (2-b)*(a-1)
a-5 = - ab+2a+b-2
ab-a-b = 3 ——> 4
Solving 3 & 4, we get
-2a = -10
a = 5
Substitute a=5 in equation 4,
5b - 5 - b = 3
4b = 8
b = 2
Therefore a = 5, b = 2 ——> Answer
A(a, 2) , B(1,3) & C(5,b) are collinear.
We can say that when the points are collinear, slope are equal.
Slope between AB = Slope between BC = Slope between AC
(2-3)/(a-1) = (3-b)/(1-5) = (2-b)/(a-5)
1/(a-1) = (3-b)/(-4) = (2-b)/(a-5)
1/(a-1) = (b-3)/(4) = (2-b)/(a-5) ——> 1
1/(a-1) = (b-3)/(4)
4 = (a-1)*(b-3)
4 = ab -3a - b + 3
ab -3a - b = 1 ——> 2
(b-3)/(4) = (2-b)/(a-5)
(b-3)* (a-5) = (2-b)*4
ab - 3a -5b + 15 = 8 - 4b
ab - 3a - b = -7 ——> 3
1/(a-1) = (2-b)/(a-5)
(a-5) = (2-b)*(a-1)
a-5 = - ab+2a+b-2
ab-a-b = 3 ——> 4
Solving 3 & 4, we get
-2a = -10
a = 5
Substitute a=5 in equation 4,
5b - 5 - b = 3
4b = 8
b = 2
Therefore a = 5, b = 2 ——> Answer
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Answer:
Correct option is
A
Step-by-step explanation:
21
A (3,4) and B(7,7) and C(x,y) or C'(x,y) are collinear points.
AB = √[(7-3)²+(7-4)²] = 5
AC = 10, given.
Slope of AC = slope of AB = (7-4)/(7-3) = 3/4
=> (y-4)/(x-3) = 3/4 --- (1)
=> 4 y - 3 x = 7 --- (2)
AC² = 10² = (y - 4)² + (x - 3)²
= [ 3/4 * (x - 3) ]² + (x-3)²
= (x-3)² * 25/16
=> x - 3 = + 8
=> x = +11 or -5
=> y = (7+3x)/4
= 10 or -2
C = (11, 10) and C' = (-5, -2)
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