The striaght lines whose direction cosines satisfies $a l + b m + c n = 0, f m n + g n l + h l m = 0$ are perpendicular or parallel if

\frac{f}{a} + \frac{g}{b} + \frac{h}{c} = 0

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

\sqrt{c h} + \sqrt{b g} + \sqrt{a f} = 0

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

\sqrt{c h} - \sqrt{b g} - \sqrt{a f} = 0

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

l_{1} l_{2} = m_{1} m_{2} = n_{1} n_{2} = 0

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct options are
A $\frac{f}{a} + \frac{g}{b} + \frac{h}{c} = 0$
B $\sqrt{c h} + \sqrt{b g} + \sqrt{a f} = 0$
C $\sqrt{c h} - \sqrt{b g} - \sqrt{a f} = 0$
Eliminating $l$ between the given relations
$f m n + (g n + h m) (\frac{b m + c n}{- a}) = 0 \Rightarrow b h {(\frac{m}{n})}^{2} + (c h + b g - a f) (\frac{m}{n}) + c g = 0 \dots (i)$
Let the direction cosines of the two lines be $l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}, n_{2}$

Assuming the roots of the above equation
$(\frac{m_{1}}{n_{1}})$ and $(\frac{m_{2}}{n_{2}})$

When the lines will be parallel then the given equation will have equal roots,
$D = 0 (c h + b g - a f)^{2} = 4 b c g h \Rightarrow (c h + b g - a f) = \pm 2 \sqrt{b c g h} \Rightarrow a f = \mp 2 \sqrt{b c g h} + c h + b g \Rightarrow (\sqrt{c h} \pm \sqrt{b g})^{2} = (\sqrt{a f})^{2} \Rightarrow (\sqrt{c h} \pm \sqrt{b g}) \pm (\sqrt{a f}) = 0$

Now $(\frac{m_{1}}{n_{1}}) \cdot (\frac{m_{2}}{n_{2}}) = \frac{c g}{b h}$

$\frac{m_{1} m_{2}}{g / b} = \frac{n_{1} n_{2}}{h / c} \dots (i i)$
Similarly eliminating $n$ between the given relations
$\frac{m_{1} m_{2}}{g / b} = \frac{l_{1} l_{2}}{f / a} \dots (i i i)$
Now lines are perpendicular, then
$l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = 0$
$∴ \frac{f}{a} + \frac{g}{b} + \frac{h}{c} = 0$

Suggest Corrections

Similar questions

The straight lines whose direction cosines are given by al + bm + cn = 0, fmn + gnl + hlm = 0 are perpendicular, if

a l + b m + c n = 0

f m n + g n l + h l m = 0

\frac{f}{a} + \frac{g}{b} + \frac{h}{c} = 0.

(f^{2} - b c) x + (c h - f g) y + (b g + h f) z = 0

(c h - f g) x + (g^{2} - c a) y + (a f - g h) z = 0

(b g - h f) x + (a f - g h) y + (h^{2} - a b) z = 0

a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 0

a l + b m + c n = 0

f m n + g l n + h l m = 0

f, g, h, a, b, c

l, m, n

f = g = h = 1

Join BYJU'S Learning Program