The coordinates of the point in which the line joining the points (2, 5, -7) and (-3, -1, 8) are intersected by the y-z plane are

Options

A. `(0, 13/5, -1)`
B. `(0, (-13)/5, -2)`
C. `(0, (-13)/5, 2/5)`
D. `(0, 13/5, 2/5)`

Answer

To find the coordinates of the point where the line joining the points \( P(2, 5, -7) \) and \( Q(-3, -1, 8) \) intersects the y-z plane, we can follow these steps:

### Step 1: Understand the y-z Plane
The equation of the y-z plane is given by \( x = 0 \). This means that at the intersection point, the x-coordinate will be 0.

**Hint:** Remember that the y-z plane is defined by all points where the x-coordinate is zero.

### Step 2: Find the Direction Ratios
The direction ratios of the line joining points \( P \) and \( Q \) can be calculated as follows:
- The direction ratios are given by \( (x_2 - x_1, y_2 - y_1, z_2 - z_1) \).
- Here, \( P(2, 5, -7) \) and \( Q(-3, -1, 8) \).

Calculating the direction ratios:
\[ x_2 - x_1 = -3 - 2 = -5 \]
\[ y_2 - y_1 = -1 - 5 = -6 \]
\[ z_2 - z_1 = 8 - (-7) = 15 \]
Thus, the direction ratios are \( (-5, -6, 15) \).

**Hint:** Direction ratios help us determine the slope of the line in three-dimensional space.

### Step 3: Parametric Equations of the Line
Using the point \( P(2, 5, -7) \) and the direction ratios, we can write the parametric equations of the line:
\[ x = 2 - 5t \]
\[ y = 5 - 6t \]
\[ z = -7 + 15t \]
where \( t \) is a parameter.

**Hint:** Parametric equations express the coordinates of points on the line in terms of a single variable.

### Step 4: Set \( x = 0 \) to Find Intersection
To find the intersection with the y-z plane, we set \( x = 0 \):
\[ 0 = 2 - 5t \]
Solving for \( t \):
\[ 5t = 2 \implies t = \frac{2}{5} \]

**Hint:** Setting \( x = 0 \) allows us to find the specific value of \( t \) at which the line intersects the y-z plane.

### Step 5: Substitute \( t \) Back into the Parametric Equations
Now, substitute \( t = \frac{2}{5} \) into the equations for \( y \) and \( z \):
\[ y = 5 - 6\left(\frac{2}{5}\right) = 5 - \frac{12}{5} = \frac{25}{5} - \frac{12}{5} = \frac{13}{5} \]
\[ z = -7 + 15\left(\frac{2}{5}\right) = -7 + 6 = -1 \]

**Hint:** Substituting back gives us the coordinates of the intersection point.

### Step 6: Write the Final Coordinates
The coordinates of the point where the line intersects the y-z plane are:
\[ (0, \frac{13}{5}, -1) \]

Thus, the final answer is:
\[ \boxed{(0, \frac{13}{5}, -1)} \]