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If the points (2,−3)(2, -3), (λ,−1)(\lambda, -1) and (0,4)(0, 4) are collinear, then the value of λ\lambda is

Solution

✅ Correct Option: 4

Given points: (2,−3)(2, -3), (λ,−1)(\lambda, -1), and (0,4)(0, 4)

For three points to be collinear, the slope between any two pairs of points must be equal.


Using points (2,−3)(2, -3) and (0,4)(0, 4):

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

m=4−(−3)0−2m = \frac{4 - (-3)}{0 - 2}

m=7−2m = \frac{7}{-2}

m=−72m = -\frac{7}{2}


Using points (2,−3)(2, -3) and (λ,−1)(\lambda, -1):

m=−1−(−3)λ−2m = \frac{-1 - (-3)}{\lambda - 2}

m=2λ−2m = \frac{2}{\lambda - 2}


Since the points are collinear, the slopes must be equal:

2λ−2=−72\frac{2}{\lambda - 2} = -\frac{7}{2}

2×2=−7×(λ−2)2 \times 2 = -7 \times (\lambda - 2)

4=−7λ+144 = -7\lambda + 14

7λ=14−47\lambda = 14 - 4

7λ=107\lambda = 10

λ=107\lambda = \frac{10}{7}


Therefore, λ=107\lambda = \frac{10}{7}

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