Question

The co-ordinates of the foot of the perpendicular drawn from the origin to the 2x + y - 2z = 18 are

• A. (4,2,-4)
• B. (1,2,-3)
• C. (4,2,4)
• D. (4,-2,-4)

Answer 1

Answer: (A)

(4,2,-4)

Answer 2

To find the foot of the perpendicular from the origin to the plane $2x + y - 2z = 18$, we use the formula for the foot of the perpendicular from a point $P(x_1, y_1, z_1)$ to a plane $ax + by + cz + d = 0$, which is given by:

$$F = \left(x_1 - \frac{a(ax_1+by_1+cz_1+d)}{a^2+b^2+c^2}, y_1 - \frac{b(ax_1+by_1+cz_1+d)}{a^2+b^2+c^2}, z_1 - \frac{c(ax_1+by_1+cz_1+d)}{a^2+b^2+c^2}\right)$$

In this case, $P = (0, 0, 0)$, $a = 2$, $b = 1$, $c = -2$, and $d = -18$. Plugging these values into the formula, we get:

$$ax_1 + by_1 + cz_1 + d = 2(0) + 1(0) - 2(0) - 18 = -18$$ $$a^2 + b^2 + c^2 = 2^2 + 1^2 + (-2)^2 = 4 + 1 + 4 = 9$$

Therefore, the foot of the perpendicular $F$ is:

$$F = \left(0 - \frac{2(-18)}{9}, 0 - \frac{1(-18)}{9}, 0 - \frac{-2(-18)}{9}\right) = \left(\frac{36}{9}, \frac{18}{9}, -\frac{36}{9}\right) = (4, 2, -4)$$

Thus, the coordinates of the foot of the perpendicular are $(4, 2, -4)$.