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Physics Guru is an online education and competitive examinations preparation portal founded by Manish Verma, an IIT Madras alumnus, providing learning resources for competitive examinations.

The emphasis is on conceptual understanding, analytical thinking and problem solving rather than simply memorising formulas and standard methods.

The objective is to help students understand what they are learning and develop the ability to solve problems they have not seen before.

Let P be a plane lx + my + nz = 0 containing the line ....

Let P be a plane lx + my + nz = 0 containing the line, $\frac{{1 - x}}{1} = \frac{{y + 4}}{2} = \frac{{z + 2}}{3}$. If plane P divides the line segment AB joining points A (-3, -6, 1) and B (2, 4, -3) in ratio k : 1 then the value of k is equal to:

(A) 4
(B) 2
(C) 1.5
(D) 3

Solution

The plane lx + my + nz = 0 contains the line $\frac{{x - 1}}{{ - 1}} = \frac{{y + 4}}{2} = \frac{{z + 2}}{3}$ having direction ratios $\equiv $ -1, 2, 3. Thus,

-l + 2m + 3n = 0

Since the line passes through (1, -4, -2) the plane would also contain this point.

So, l - 4m -2n = 0

Thus, l = 8m & n = 2m

The equation of plane can be written as, 8mx + my + 2mz = 0 or 8x + y + 2z = 0.

$P \equiv \left( {\frac{{ - 3 + 2k}}{{k + 1}},\frac{{ - 6 + 4k}}{{k + 1}},\frac{{1 - 3k}}{{k + 1}}} \right)$ lies on the plane.

$\therefore 8\left( {\frac{{ - 3 + 2k}}{{k + 1}}} \right) + \frac{{ - 6 + 4k}}{{k + 1}} + 2\left( {\frac{{1 - 3k}}{{k + 1}}} \right) = 0$

$ \Rightarrow 8( - 3 + 2k) + ( - 6 + 4k) + 2(1 - 3k) = 0$

$ \Rightarrow k = 2$

Answer: (B)