Question

A line drawn from a point $A(-2,-2,3)$ and parallel to the line $\frac{x}{-2} = \frac{y}{2} = \frac{z}{-1}$ meets the $YOZ$ plane in point $P$, then the coordinates of the point $P$ are

• A. $(0, 4, -4)$
• B. $(0, 2, 2)$
• C. $(0, -2, 2)$
• D. $(0, -4, 4)$

Hint:

Any point on the $YOZ$ plane must have an $x$-coordinate of 0, which all options correctly show. Next, examine the direction ratios: for every change of $+2$ units along $x$ (moving from $-2$ to $0$), $y$ must change by $-2$ units (based on the ratio $\frac{\Delta x}{-2} = \frac{\Delta y}{2}$). Thus, $y = -2 - 2 = -4$, isolating option (D) instantly without full calculation!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We need to find the coordinates of a point $P$. This point is formed by the intersection of a specific straight line with the $YOZ$ plane. The line passes through $A(-2,-2,3)$ and runs parallel to a given reference line whose direction ratios are $(-2, 2, -1)$.

Step 2: Key Formula or Approach:
The equation of a line passing through point $(x_1, y_1, z_1)$ with direction ratios $(l, m, n)$ is given by: $$\frac{x - x_1}{l} = \frac{y - y_1}{m} = \frac{z - z_1}{n}$$ Since our target line is parallel to the reference line, it inherits the exact same direction ratios: $(-2, 2, -1)$. Any point on this line can be written in terms of a scalar parameter $\lambda$. Since it intersects the $YOZ$ plane, the $x$-coordinate of point $P$ must be exactly equal to 0.

Step 3: Detailed Explanation:
Let's write down the symmetric equation of our line: $$\frac{x - (-2)}{-2} = \frac{y - (-2)}{2} = \frac{z - 3}{-1}$$ $$\frac{x + 2}{-2} = \frac{y + 2}{2} = \frac{z - 3}{-1} = \lambda$$ Express the general coordinates of any point on this line in terms of $\lambda$: $$x = -2\lambda - 2$$ $$y = 2\lambda - 2$$ $$z = -\lambda + 3$$ Since point $P$ lies on the $YOZ$ plane, its $x$-coordinate must be 0: $$-2\lambda - 2 = 0 \implies -2\lambda = 2 \implies \lambda = -1$$ Now, substitute $\lambda = -1$ back into our equations to compute the $y$ and $z$ coordinates of point $P$: $$y = 2(-1) - 2 = -2 - 2 = -4$$ $$z = -(-1) + 3 = 1 + 3 = 4$$ Combining these components, the coordinates of point $P$ are $(0, -4, 4)$.

Step 4: Final Answer:
The coordinates of point $P$ are $(0, -4, 4)$, which corresponds to option (D).