If the lines x-12 = y-3-3 = z-3 and x-45 = y-12 = z intersect, then the magnitude of the minimum value of 8 is:

Given: The lines are x-12 = y-3-3 = z-3 and x-45 = y-12 = z. Find: The magnitude of the minimum value of 8. The given lines are in symmetric form. Two lines intersect if the coplanarity condition of their direction vectors and the vector joining points on them is satisfied. Using the determinant condition: vmatrix 2 & -3 & 5 & 2 & 4-1 & 1-3 & 0-3 vmatrix = 0 Expanding the determinant as shown: 2 (2 (-3)- (-2))-(-3) (5 (-3)- 3)+ (5 (-2)-2 3)=0 From the solution, this gives a relation between and. Using optimization, the minimum value of 8 is obtained as 18. Therefore, the magnitude of the minimum value is 18.

If the lines x-12 = y-3-3 = z-3 and x-45 = y-12 = z intersect, th… — Solution | JEE Mathematics

NVAMediumJEE 2023Equation of Line in 3D

If the lines $\frac{x-1}{2} = \frac{y-3}{-3} = \frac{z-3}{\alpha}$ and $\frac{x-4}{5} = \frac{y-1}{2} = \frac{z}{\beta}$ intersect, then the magnitude of the minimum value of $8\alpha\beta$ is :

Answer

Correct answer:18

Step-by-step solution

Standard Method

Given: The lines are $\frac{x-1}{2} = \frac{y-3}{-3} = \frac{z-3}{\alpha}$ and $\frac{x-4}{5} = \frac{y-1}{2} = \frac{z}{\beta}$ .

Find: The magnitude of the minimum value of $8\alpha\beta$ .

The given lines are in symmetric form. Two lines intersect if the coplanarity condition of their direction vectors and the vector joining points on them is satisfied.

Using the determinant condition:

\begin{vmatrix} 2 & -3 & \alpha \\ 5 & 2 & \beta \\ 4-1 & 1-3 & 0-3 \end{vmatrix} = 0

Determinant Expansion

Expanding the determinant as shown:

2\left(2\cdot(-3)-\beta\cdot(-2)\right)-(-3)\left(5\cdot(-3)-\beta\cdot3\right)+\alpha\left(5\cdot(-2)-2\cdot3\right)=0

From the solution, this gives a relation between $\alpha$ and $\beta$ . Using optimization, the minimum value of $8\alpha\beta$ is obtained as $18$ .

Therefore, the magnitude of the minimum value is $18$ .

Common mistakes

Using the wrong joining vector in the coplanarity determinant. The vector must be formed from one point on the first line to one point on the second line. A wrong sign changes the relation between $\alpha$ and $\beta$ . Use coordinates carefully before expanding the determinant.
Assuming that intersecting lines only need proportional direction ratios. That condition is for parallel lines, not intersecting skew-check problems. Here, use the determinant condition for coplanarity together with the given symmetric forms.
Minimizing $\alpha\beta$ without using the relation obtained from the intersection condition. The optimization step is valid only after expressing one variable in terms of the other from the determinant equation.

Practice more Equation of Line in 3D questions

Get unlimited AI-adaptive practice, mastery tracking, and an AI tutor that explains every step — free to start.

Start free trial More on Equation of Line in 3D

Answer

Step-by-step solution

Common mistakes

Practice more Equation of Line in 3D questions

Related questions