Question

Show that the straight lines whose direction cosines are given by the equations `a l+b m+c n=0a n dul^2+z m^2=v n^2+w n^2=0` are parallel or perpendicular as `(a^2)/u+(b^2)/v+(c^2)/w=0ora^2(v+w)+b^2(w+u)+c^2(u+v)=0.`

Answer 1

Here, `l=-((bm+cn))/(a)and"ul"^(2)+m^(2)v+wn^(2)=0`.
Elimiating l, we get
`(u(bm+cn)^(2))/(a^(2))+vm^(2)+wn^(2)=0`
`u(b^(2)m^(2)+2bcmn+c^(2)n^(2))+va^(2)m^(2)+wa^(2)n^(2)=0`
`(b^(2)u+a^(2)v)m^(2)+(2bcu)m+(c^(2)u+a^(2)w)n^(2)=0`
or `(b^(2)u+a^(2)v)((m)/(n))^(2)+(2bcu)((m)/(n))`
`+(c^(2)u+a^(2)w)=0` which is quadratic in (m/n) having roots `m_(1)//n_(1)andm_(2)//n_(2)`
a. If the straight lines are parallel, the quadratic in m/n has equal roots, i.e., discriminant =0
`implies(2bcu)^(2)-4(b^(2)u+a^(2)v)(c^(2)u+a^(2)w)=`
or `b^(2)c^(2)u^(2)=(b^(2)u+a^(2)v)(c^(2)u+a^(2)w)`
or `a^(2)vw+b^(2)uw+c^(2)uv=0`
or `a^(2)/(u)+b^(2)/(v)+c^(2)/(w)=0`
b. If the straight lines are perpendicular, then
`(m_(1))/(n_(1))=(m_(2))/(n_(2))=(c^(2)u+a^(2)w)/(b^(2)u+a^(2)v )` (product of roots)
or `(m_(1)m_(2))/(c^(2)u+a^(2)w)=(n_(1)n_(2))/(b^(2)u+a^(2)w)" "(i)`
Similarly, by elimingting n, we get
`(l_(1)l_(2))/(b^(2)w+c^(2)v)=(m_(1)m_(2))/(c^(2)u+a^(2)w)" "(ii)`
From (i) and (ii),
`(l_(1)l_(2))/(b^(2)w+c^(2)v)=(m_(1)m_(2))/(c^(2)u+a^(2)w)=(n_(1)n_(2))/(b^(2)u+a^(2)v)=lamda`
Since they are perpendicular,
`l_(1)l_(2)+m_(1)m_(2)+n_(1)n_(2)=0`
or `lamda(b^(2)w+c^(2)v)+lamda(c^(2)u+a^(2)w)+lamda(b^(2)u+a^(2)v)=0`
or `a^(2)(v+w)+b^(2)(w+u)+c^(2)(u+v)=0`