Question

The straight line L1 passes through the points with coordinates (4, 6) and (12, 2) The straight line L2 passes through the origin and has gradient -3. The lines L1 and L2 intersect at point P. Find the coordinates of P.

Answer 1

To find the intersection point P of lines L1 and L2, we calculate equations for each line, and then solve for their common solution. The coordinates of point P are (4, -12).

Step-by-step explanation:

To find the coordinates of point P where line L1 (passing through the points (4, 6) and (12, 2)) intersects with line L2 (passing through the origin with a gradient of -3), we first need to determine the equation of each line.

For L1, we can find the slope (m) using the formula: m = (y2 - y1) / (x2 - x1) = (2 - 6) / (12 - 4) = -4 / 8 = -0.5. Using one of the points and the slope, we can apply the point-slope form to find L1's equation: y - y1 = m(x - x1). Substituting the slope and point (4, 6), we get: y - 6 = -0.5(x - 4).

For L2, since the slope is given as -3 and it passes through the origin, its equation is simply y = -3x.

Now, we can equate the two equations to find the intersection point P. So,
-3x = -0.5x + (6 - 0.5(4)). Solving for x, we get x = 4. Substituting this x value back into either equation, we can find y, resulting in y = -3(4) = -12. Therefore, point P is at (4, -12).