If lines frac{x-1}{2}=frac{y+1}{3}=frac{z-1}{ | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Two lines $\frac{x-x_1}{a_1} = \frac{y-y_1}{b_1} = \frac{z-z_1}{c_1}$ and $\frac{x-x_2}{a_2} = \frac{y-y_2}{b_2} = \frac{z-z_2}{c_2}$ intersect if

Vedclass · Accepted Answer

$\frac{9}{2}$

Vedclass · Answer

(A) Two lines $\frac{x-x_1}{a_1} = \frac{y-y_1}{b_1} = \frac{z-z_1}{c_1}$ and $\frac{x-x_2}{a_2} = \frac{y-y_2}{b_2} = \frac{z-z_2}{c_2}$ intersect if and only if the determinant of the matrix formed by the differences of their points and their direction ratios is zero: $\left|\begin{array}{ccc} x_2-x_1 & y_2-y_1 & z_2-z_1 \ a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \end{array}ight| = 0$ Given the lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{2}=\frac{z-0}{1}$,we have $(x_1, y_1, z_1) = (1, -1, 1)$ and $(x_2, y_2, z_2) = (3, k, 0)$. The direction ratios are $(a_1, b_1, c_1) = (2, 3, 4)$ and $(a_2, b_2, c_2) = (1, 2, 1)$. Substituting these into the determinant condition: $\left|\begin{array}{ccc} 3-1 & k-(-1) & 0-1 \ 2 & 3 & 4 \ 1 & 2 & 1 \end{array}ight| = 0$ $\Rightarrow \left|\begin{array}{ccc} 2 & k+1 & -1 \ 2 & 3 & 4 \ 1 & 2 & 1 \end{array}ight| = 0$ Expanding along the first row: $2(3(1) - 4(2)) - (k+1)(2(1) - 4(1)) - 1(2(2) - 3(1)) = 0$ $2(3-8) - (k+1)(2-4) - 1(4-3) = 0$ $2(-5) - (k+1)(-2) - 1(1) = 0$ $-10 + 2k + 2 - 1 = 0$ $2k - 9 = 0$ $2k = 9$ $k = \frac{9}{2}$

If lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{2}=\frac{z-0}{1}$ intersect,then the value of $k$ is

Similar Questions

The angle between the two lines $\frac{x-4}{1}=\frac{y+4}{2}=\frac{z+1}{2}$ and $\frac{x+1}{2}=\frac{y+3}{2}=\frac{z-4}{-1}$ is

If the length of the perpendicular from the point $P(\beta, 0, \beta) \, (\beta \neq 0)$ to the line $\frac{x}{1} = \frac{y - 1}{0} = \frac{z + 1}{-1}$ is $\sqrt{\frac{3}{2}}$,then $\beta$ is equal to

The square of the distance of the point $P(5,6,7)$ from the line $\frac{x-2}{2} = \frac{y-5}{3} = \frac{z-2}{4}$ is equal to:

Find the angle between the pair of lines $\frac{x+3}{3}=\frac{y-1}{5}=\frac{z+3}{4}$ and $\frac{x+1}{1}=\frac{y-4}{1}=\frac{z-5}{2}$.

Show that the straight lines whose direction cosines are given by $2l + 2m - n = 0$ and $mn + nl + lm = 0$ are at right angles.

Vedclass Test Series

Exam Paper Generator

Online Exam Module