Question

⇢ Solve⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀​

Answer 1

Answer:

the value is 1.

Step-by-step explanation:

I hope it help you

Answer 2

$\large\underline{\sf{Solution-}}$

Given that, the lines

$\rm :\longmapsto\:\dfrac{x - 1}{2} = \dfrac{y + 1}{3} = \dfrac{z - 1}{4}$

and

$\rm :\longmapsto\:\dfrac{x - 3}{1} = \dfrac{y - k}{2} = \dfrac{z}{1}$

are intersecting.

★ Since, lines are intersecting, it means lines are coplanar.

We know, two lines are coplanar if

$\rm \: \: \: \: \begin{gathered}\sf \left | \begin{array}{ccc}x_2 - x_1&y_2 - y_1&z_2 - z_1\\a_1&b_1& c_1\\a_2& b_2& c_2\end{array}\right | \end{gathered} = 0$

Here,

$\rm :\longmapsto\:x_1 = 1 \\ \rm :\longmapsto\:y_1 = - 1 \\ \rm :\longmapsto\:z_1 = 1 \\ \rm :\longmapsto\:x_2 = 3 \\ \rm :\longmapsto\:y_2 = k \\ \rm :\longmapsto\:z_2 = 0 \\ \rm :\longmapsto\:a_1 = 2 \\ \rm :\longmapsto\:b_1 = 3 \\ \rm :\longmapsto\:c_1 = 4 \\ \rm :\longmapsto\:a_2 = 1 \\ \rm :\longmapsto\:b_2 = 2 \\ \rm :\longmapsto\:c_2 = 1$

★ So on substituting the values,

$\rm \: \: \: \: \begin{gathered}\sf \left | \begin{array}{ccc} 3 - 1& k+1&0 - 1\\2&3& 4\\1& 2& 1\end{array}\right | \end{gathered} = 0$

$\rm \: \: \: \: \begin{gathered}\sf \left | \begin{array}{ccc} 2& k+1& - 1\\2&3& 4\\1& 2& 1\end{array}\right | \end{gathered} = 0$

$\rm :\longmapsto\:2(3 - 8) - (k + 1)(2 - 4) - 1(4 - 3) = 0$

$\rm :\longmapsto\:2( - 5) - (k + 1)( - 2) - 1(1) = 0$

$\rm :\longmapsto\: - 10 + 2k + 2 - 1 = 0$

$\rm :\longmapsto\: - 9 + 2k = 0$

$\bf\implies \:k = \dfrac{9}{2}$

Now,

★ Equation of plane contains these lines is

$\rm \: \: \: \: \begin{gathered}\sf \left | \begin{array}{ccc}x - x_1&y - y_1&z - z_1\\a_1&b_1& c_1\\a_2& b_2& c_2\end{array}\right | \end{gathered} = 0$

$\rm \: \: \: \: \begin{gathered}\sf \left | \begin{array}{ccc} x - 1& y+1&z - 1\\2&3& 4\\1& 2& 1\end{array}\right | \end{gathered} = 0$

$\rm :\longmapsto\:(x - 1)(3 - 8) - (y + 1)(2 - 4) + (z - 1)(4 - 3) = 0$

$\rm :\longmapsto\:(x - 1)( - 5) - (y + 1)( - 2) + (z - 1)(1) = 0$

$\rm :\longmapsto\: - 5x + 5 + 2y + 2 + z - 1 = 0$

$\bf\implies \:5x - 2y - z - 6 = 0$