Question

If the line joining the points \((-1,2,5)\) and \((3,4,-10)\) intersects the \(xy\)-plane at the point \((x, y, z)\), then \(\dfrac{y}{x}\) is equal to ________.

Hint:

Write the line in parametric form and put \(z = 0\), because every point of the \(xy\)-plane has \(z = 0\).

Answer 1

Correct Answer: 8

Let \(A(-1,2,5)\) and \(B(3,4,-10)\). The direction ratios of \(AB\) are \((3+1,\ 4-2,\ -10-5) = (4,2,-15)\).
A general point of the line is \((-1+4t,\ 2+2t,\ 5-15t)\).
On the \(xy\)-plane \(z = 0\), so \(5-15t = 0\), which gives \(t = \dfrac{1}{3}\).
Then \(x = -1+\dfrac{4}{3} = \dfrac{1}{3}\) and \(y = 2+\dfrac{2}{3} = \dfrac{8}{3}\).
Therefore \(\dfrac{y}{x} = \dfrac{8/3}{1/3} = 8\).
Answer: 8