Finding the Length of a Line Segment Intersecting Axes | Geometry Problem

To find the x-intercept, we set y = 0 in the equation of the line:

3x + ky + 12 = 0 3x + (-4)(0) + 12 = 0 3x + 12 = 0 3x = -12 x = -4

So the x-intercept P is (-4, 0).

To find the y-intercept, we set x = 0 in the equation of the line:

3x + ky + 12 = 0 3(0) + ky + 12 = 0 ky + 12 = 0 ky = -12 y = -12/k

When k = -4, y = -12/(-4) = 3. So the y-intercept Q is (0, 3).

To find the length of PQ, we use the distance formula:

PQ = sqrt((x2 - x1)^2 + (y2 - y1)^2) PQ = sqrt((0 - (-4))^2 + (3 - 0)^2) PQ = sqrt((4)^2 + (3)^2) PQ = sqrt(16 + 9) PQ = sqrt(25) PQ = 5

Therefore, when k = -4, the length of PQ is 5 units.