Question

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

• A. \(\frac{1}{3}\)
• B. 3
• C. \(\frac{7}{10}\)
• D. \(\frac{10}{7}\)

Answer 1

The Correct Option is D

To find the value of λ for which the points (2, -3), (λ, -1), and (0, 4) are collinear, we use the condition for collinearity that states if three points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) are collinear, then the area of the triangle formed by these points is zero. The formula for the area of a triangle given three points is:

\(\text{Area}=\frac{1}{2}\left|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)\right|\)

Substituting the given points (2, -3), (λ, -1), and (0, 4) into the formula:

\(\text{Area}=\frac{1}{2}\left|2(-1-4) + λ(4+3) + 0(-3+1)\right|\)

Simplifying further:

\(\frac{1}{2}\left|2(-5) + λ(7) + 0\right|=\frac{1}{2}\left|-10 + 7λ\right|=0\)

For the area to be zero, the expression inside the absolute value must be zero:

\(-10 + 7λ = 0\)

Solving for λ:

\(7λ = 10\)

\(λ = \frac{10}{7}\)

Thus, the value of λ that makes the points collinear is \(\frac{10}{7}\).