Question

Let two points be A(1,-1) and B(0,2) .If a point P(h,k) be such that the area of ∆PAB is 5 sq units and it lies on the line 3x+y-4 λ=0 ,then the value of λ is

A. 4
B. 1
C. -3
D. 3

Answer 1

To find the value of λ, we substitute the values of h and k in the equation of the line and use the formula for the area of a triangle. Simplifying the equation, we find two possibilities: -h + 2k = 9 or h - 4k = -21. The correct value of λ is 3.

Step-by-step explanation:

To find the value of λ, we first need to find the coordinates of the point P(h, k) that lies on the line 3x + y - 4λ = 0 and also forms a triangle with points A(1, -1) and B(0, 2) with an area of 5 square units.

First, we substitute the value of h and k in the equation of the line to find the values of λ. We get 3h + k - 4λ = 0.

Then, we use the formula for the area of a triangle: Area = 1/2 * base * height. Plugging in the coordinates of points A, B, and P, we get 5 = 1/2 * |(1-0)(k-(-1)) - (h-0)(2-(-1))|.

Simplifying the equation, we get |4h + 3k + 2λ - 2| = 10.

Now, we substitute the value of λ from the equation of the line into the equation for the area of the triangle. We get |4h + 3k - 3h - k - 8| = 10.

Solving this equation, we find two possibilities: -h + 2k = 9 or h - 4k = -21. These equations represent the two lines that pass through the point P and are perpendicular to the line 3x + y - 4λ = 0.

The value of λ can be found by substituting the values of h and k into the equation of the line. The correct value of λ is 3.