Question

Find the co-ordinates of the point on line \( x = \frac{y-1}{2} = \frac{z-2}{3} \) whose \( y \) coordinate is 3 times the \( x \) coordinate.

Hint:

Using a parameter \( \lambda \) is the standard way to find specific points on a line given geometric constraints.
Always verify your final answer by checking if the point satisfies both the line equation and the given condition.
\( \frac{1}{1} = \frac{3-1}{2} = \frac{5-2}{3} \implies 1 = 1 = 1 \), and \( 3 = 3(1) \). Correct.

Answer 1

Concept:
• The symmetrical form of a line in 3D is \( \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} \), where \( (x_1, y_1, z_1) \) is a point on the line and \( (a, b, c) \) are the direction ratios.
• A general point on this line can be represented by equating the expression to a constant parameter \( \lambda \).
• By solving for \( x, y, \) and \( z \) in terms of \( \lambda \), we can apply any given conditions to find the specific point.

Step 1: Express the coordinates of a general point on the line in terms of a parameter
Let the given equation of the line be set equal to a parameter \( \lambda \):
\[ x = \frac{y-1}{2} = \frac{z-2}{3} = \lambda \]
From this, we can write the individual coordinates as:
\[ x = \lambda \]
\[ \frac{y-1}{2} = \lambda \implies y = 2\lambda + 1 \]
\[ \frac{z-2}{3} = \lambda \implies z = 3\lambda + 2 \]
So, any general point \( P \) on the line has coordinates \( (\lambda, 2\lambda + 1, 3\lambda + 2) \).

Step 2: Apply the given condition to find the value of the parameter
The problem states that the \( y \) coordinate is 3 times the \( x \) coordinate.
Mathematically, this means:
\[ y = 3x \]
Substitute the expressions for \( x \) and \( y \) from
Step 1:
\[ 2\lambda + 1 = 3(\lambda) \]
\[ 2\lambda + 1 = 3\lambda \]
Subtract \( 2\lambda \) from both sides:
\[ 1 = \lambda \]
\[ \lambda = 1 \]

Step 3: Determine the coordinates of the specific point
Now, substitute \( \lambda = 1 \) back into the expressions for the coordinates:
\[ x = \lambda = 1 \]
\[ y = 2(1) + 1 = 3 \]
\[ z = 3(1) + 2 = 5 \]
Therefore, the required coordinates are \( (1, 3, 5) \).