Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

The equations of the given lines can be rewritten in standard symmetric form as:

$\frac{x-3}{-2}=\frac{y-\frac{2}{5}}{\frac{3\lambda +1}{5}}=\frac{z-5}{-1}$

and

$\frac{x+2}{-1}=\frac{y-\frac{1}{3}}{-\frac{7}{3}}=\frac{z-4}{-2\mu }$

The direction ratios of the first line are $a_1=-2$, $b_1=\frac{3\lambda +1}{5}$, $c_1=-1$.

The direction ratios of the second line are $a_2=-1$, $b_2=-\frac{7}{3}$, $c_2=-2\mu$.

Since the lines are at right angles, the dot product of their direction vectors is zero:

$a_1{a}_2+b_1{b}_2+c_1{c}_2=0$

$(-2)(-1)+\left(\frac{3\lambda +1}{5}\right)\left(-\frac{7}{3}\right)+(-1)(-2\mu )=0$

$2-\frac{7(3\lambda +1)}{15}+2\mu =0$

Multiplying the entire equation by $15$ gives:

$30-7(3\lambda +1)+30\mu =0$

$30-21\lambda -7+30\mu =0$

$23-21\lambda +30\mu =0$

$21\lambda -30\mu =23$

$3(7\lambda -10\mu )=23$

$7\lambda -10\mu =\frac{23}{3}$

Answer: $\frac{23}{3}$