Class CBSE Class 12 Three Dimensional Geometry Question

Question

If P is a point on the line segment joining (3, 6, -1) and (6, 2, -2) and y-coordinate of P is 4, then its z-coordinate is:

Accepted Answer

Correct Answer: A. Explanation:

We are given two points, $A = (3, 6, -1)$ and $B = (6, 2, -2)$. The point $P$ lies on the line segment $AB$, and its $y$-coordinate is $4$. We need to find its $z$-coordinate.

Let $P$ divide the line segment $AB$ in the ratio $\lambda:1$.

Step 1: Find the Ratio ($\lambda$)

We use the section formula for the $y$-coordinate, where $y=4$, $y_1=6$, and $y_2=2$:

$$y = \frac{\lambda y_2 + y_1}{\lambda + 1}$$ $$4 = \frac{\lambda(2) + 6}{\lambda + 1}$$ $$4(\lambda + 1) = 2\lambda + 6$$ $$4\lambda + 4 = 2\lambda + 6$$ $$2\lambda = 2$$ $$\mathbf{\lambda = 1}$$

The point $P$ is the **midpoint** of the segment $AB$ since $\lambda = 1$.

Step 2: Find the $z$-coordinate ($z$)

Now, we use the section formula for the $z$-coordinate with $\lambda=1$, $z_1=-1$, and $z_2=-2$:

$$z = \frac{\lambda z_2 + z_1}{\lambda + 1}$$ $$z = \frac{(1)(-2) + (-1)}{1 + 1}$$ $$z = \frac{-2 - 1}{2}$$ $$\mathbf{z = -\frac{3}{2}}$$

Step 1: Find the Ratio (\(\lambda\))

Step 2: Find the \(z\)-coordinate (\(z\))

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