Find the Length of Line Segment PQ on Line L: 3x + ky + 12 = 0

To find the length of PQ, we need to find the coordinates of points P and Q.

First, let's find the x-coordinate of point P when L cuts the x-axis. We know that when a line cuts the x-axis, the y-coordinate is 0. So, we substitute y = 0 into the equation of L and solve for x:

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

Therefore, the x-coordinate of point P is -4.

Next, let's find the y-coordinate of point Q when L cuts the y-axis. We know that when a line cuts the y-axis, the x-coordinate is 0. So, we substitute x = 0 into the equation of L and solve for y:

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

Therefore, the y-coordinate of point Q is -12/k.

Now, we can find the length of PQ using the distance formula:

Distance PQ = √((x2 - x1)^2 + (y2 - y1)^2)

Substituting the coordinates of points P and Q, we have:

Distance PQ = √((-4 - 0)^2 + (-12/k - 0)^2) = √((-4)^2 + (-12/k)^2) = √(16 + 144/k^2) = √(16k^2 + 144)/k

Therefore, the length of PQ is √(16k^2 + 144)/k.