Question

Find a point \( P \) on the line \( \frac{x + 5}{1} = \frac{y + 3}{4} = \frac{z - 6}{-9} \) such that its distance from point \( Q(2, 4, -1) \) is 7 units. Also, find the equation of the line joining \( P \) and \( Q \).

Hint:

When finding the point on a line at a specific distance from another point, use the distance formula and parametric equations. The parametric equation can be used to express the coordinates of the point on the line.

Answer 1

The equation of the line is: \[ \frac{x + 5}{1} = \frac{y + 3}{4} = \frac{z - 6}{-9} = t. \] So, the parametric equations for the line are: \[ x = -5 + t, \quad y = -3 + 4t, \quad z = 6 - 9t. \] Now, let \( P(x, y, z) \) be the point on the line. We are given that the distance between \( P \) and \( Q(2, 4, -1) \) is 7 units. The distance formula between two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}. \] Substitute the coordinates of \( P \) and \( Q \): \[ 7 = \sqrt{(x - 2)^2 + (y - 4)^2 + (z + 1)^2}. \] Substitute the parametric equations of \( P \) into this equation, and solve for \( t \). After simplifying the expression, you will get the value of \( t \), and then substitute \( t \) back into the parametric equations to find the coordinates of \( P \). Finally, find the equation of the line joining \( P(x, y, z) \) and \( Q(2, 4, -1) \). This can be done using the parametric form of the line: \[ \frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1}. \] Substitute the coordinates of \( P \) and \( Q \) to get the equation of the line joining \( P \) and \( Q \).