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Find the value of k for which the points A (3, 2), B (4, k) and C (5, 3) are collinear.
- a)5/2
- b)2/5
- c)3/5
- d)1/5
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
Find the value of k for which the points A (3, 2), B (4, k)and C (5, 3...
Explanation:
To find the value of k, we need to check whether the points A, B, and C are collinear or not.
Method 1: Using Slope
If the points are collinear, then the slope of the line passing through any two points should be equal to the slope of the line passing through the other two points.
Slope of AB = (k - 2)/(4 - 3) = k - 2
Slope of BC = (3 - k)/(5 - 4) = 3 - k
If AB and BC are collinear, then their slopes should be equal.
k - 2 = 3 - k
2k = 5
k = 5/2
Therefore, the value of k is 5/2.
Method 2: Using Area of Triangle
If the points are collinear, then the area of the triangle formed by these points should be zero.
Area of triangle ABC = (1/2) * |(3 - 4)(k - 3) - (5 - 4)(2 - k)| = 0
Simplifying the above equation, we get
(1/2) * |-k + 9 + 1 + k - 4| = 0
(1/2) * |6| = 0
As the area of the triangle is zero, the points are collinear.
Therefore, the value of k is 5/2.
Hence, the correct answer is option 'A' (5/2).
To find the value of k, we need to check whether the points A, B, and C are collinear or not.
Method 1: Using Slope
If the points are collinear, then the slope of the line passing through any two points should be equal to the slope of the line passing through the other two points.
Slope of AB = (k - 2)/(4 - 3) = k - 2
Slope of BC = (3 - k)/(5 - 4) = 3 - k
If AB and BC are collinear, then their slopes should be equal.
k - 2 = 3 - k
2k = 5
k = 5/2
Therefore, the value of k is 5/2.
Method 2: Using Area of Triangle
If the points are collinear, then the area of the triangle formed by these points should be zero.
Area of triangle ABC = (1/2) * |(3 - 4)(k - 3) - (5 - 4)(2 - k)| = 0
Simplifying the above equation, we get
(1/2) * |-k + 9 + 1 + k - 4| = 0
(1/2) * |6| = 0
As the area of the triangle is zero, the points are collinear.
Therefore, the value of k is 5/2.
Hence, the correct answer is option 'A' (5/2).
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Community Answer
Find the value of k for which the points A (3, 2), B (4, k)and C (5, 3...
The points A (3, 2), B (4, K) and C (5, 3) are collinear then area (ΔABC) = 0
⇒ 1/2 [3(k − 3)+ 4(3 − 2)+5(2 − k)] = 0
⇒ 3k - 9 + 4 + 10 - 5k = 0
⇒ -2k + 5 = 0 ⇒ k = 5/2
⇒ 1/2 [3(k − 3)+ 4(3 − 2)+5(2 − k)] = 0
⇒ 3k - 9 + 4 + 10 - 5k = 0
⇒ -2k + 5 = 0 ⇒ k = 5/2
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