Question

If a point \( P \) with \( x \)-coordinate \( 7 \) lies on the line joining the points \( A(1,2,3) \) and \( B(4,6,8) \), then the coordinates of the point \( P \) are

• A. \( (7,10,-13) \)
• B. \( (7,-10,13) \)
• C. \( (7,10,12) \)
• D. \( (7,10,13) \)
• E. \( (7,10,15) \)

Hint:

To find a point on a line joining two points, first write the parametric form using the direction vector, then use the given coordinate condition to determine the parameter.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Any point on the line passing through points A and B can be represented parametrically. We first find the equation of the line and then use the given x-coordinate to find the value of the parameter, which in turn gives the other coordinates.
Step 2: Key Formula or Approach:
1. Find the direction ratios (DRs) of the line passing through A(1, 2, 3) and B(4, 6, 8). DRs = \((x_2-x_1, y_2-y_1, z_2-z_1)\). 2. Write the symmetric equation of the line: \( \frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c} \). 3. Set this equation equal to a parameter, say \(\lambda\). Any point on the line can be written in terms of \(\lambda\). 4. Use the given x-coordinate of P to solve for \(\lambda\). 5. Substitute the value of \(\lambda\) back to find the y and z coordinates of P.
Step 3: Detailed Explanation:
The points are A(1, 2, 3) and B(4, 6, 8).
The direction ratios of the line AB are:
\(a = 4-1 = 3\)
\(b = 6-2 = 4\)
\(c = 8-3 = 5\)
The equation of the line passing through A(1, 2, 3) with these direction ratios is:
\[ \frac{x-1}{3} = \frac{y-2}{4} = \frac{z-3}{5} \] Let this be equal to a parameter \(\lambda\).
\[ \frac{x-1}{3} = \frac{y-2}{4} = \frac{z-3}{5} = \lambda \] Any point P on this line can be represented as:
\(x = 3\lambda + 1\)
\(y = 4\lambda + 2\)
\(z = 5\lambda + 3\)
We are given that the x-coordinate of P is 7.
\[ 7 = 3\lambda + 1 \] \[ 6 = 3\lambda \] \[ \lambda = 2 \] Now substitute \(\lambda=2\) back into the expressions for y and z:
\(y = 4(2) + 2 = 8 + 2 = 10\)
\(z = 5(2) + 3 = 10 + 3 = 13\)
So, the coordinates of the point P are (7, 10, 13).
Step 4: Final Answer:
The coordinates of point P are (7, 10, 13). This corresponds to option (D).