Find the equation of the line joining A(1,3) and B(0,0) using determinants and find k if D(k,0) is a point such that area of triangle ABD is 3 sq units.
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Step-by-Step Solution
Step 1: Equation of line using determinants
To find the equation of a line passing through two points (x1,y1) and (x2,y2) using determinants, we consider a general point (x, y) on the line. The area of the triangle formed by these three collinear points must be zero. The formula for the area of a triangle using determinants is given by the expression shown.
Step 2: Substitute points and solve for equation
Substitute the coordinates of points A (1,3) and B (0,0) into the determinant formula. Expanding the determinant along the first row gives us x(3⋅1−0⋅1)−y(1⋅1−0⋅1)+1(1⋅0−3⋅0)=0. Simplifying this expression yields the equation of the line.
Step 3: Area of triangle ABD
Now, we need to find the value of k such that the area of triangle ABD is 3 square units. We use the same determinant formula for the area of a triangle, substituting the coordinates of points A (1,3), B (0,0), and D (k,0).
Step 4: Substitute points and solve for k
Substitute the coordinates of A (1,3), B (0,0), and D (k,0) into the area formula. Expanding the determinant along the first row gives 1(0⋅1−0⋅1)−3(0⋅1−k⋅1)+1(0⋅0−0⋅k). This simplifies to 21∣−3(−k)∣=3, which further simplifies to 21∣3k∣=3. Solving for k gives two possible values, k=2 or k=−2.