If the direction cosines of a line L are (pq, | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) The direction cosines of a line are denoted by $(l, m, n)$. Given $(l, m, n) = (pq, q, q)$.We know that for any line,$l^2 + m^2 + n^2 = 1$.Substit

Vedclass · Accepted Answer

$16 : 9$

Vedclass · Answer

(D) The direction cosines of a line are denoted by $(l, m, n)$. Given $(l, m, n) = (pq, q, q)$.We know that for any line,$l^2 + m^2 + n^2 = 1$.Substituting the given values: $(pq)^2 + q^2 + q^2 = 1$,which simplifies to $p^2q^2 + 2q^2 = 1$,or $q^2(p^2 + 2) = 1$.Also,the direction cosine $l$ is given by $\cos(\alpha)$,where $\alpha$ is the angle with the $X$-axis.Given $\alpha = \frac{\pi}{3}$,so $l = \cos(\frac{\pi}{3}) = \frac{1}{2}$.Since $l = pq$,we have $pq = \frac{1}{2}$,which implies $p^2q^2 = \frac{1}{4}$.Substitute $p^2q^2 = \frac{1}{4}$ into the equation $p^2q^2 + 2q^2 = 1$:$\frac{1}{4} + 2q^2 = 1$$2q^2 = 1 - \frac{1}{4} = \frac{3}{4}$$q^2 = \frac{3}{8}$.Now,find $p^2$ using $p^2q^2 = \frac{1}{4}$:$p^2(\frac{3}{8}) = \frac{1}{4}$$p^2 = \frac{1}{4} 	imes \frac{8}{3} = \frac{2}{3}$.Finally,the ratio $p^2 : q^2 = \frac{2}{3} : \frac{3}{8} = \frac{2}{3} 	imes \frac{8}{3} = \frac{16}{9}$.

If the direction cosines of a line $L$ are $(pq, q, q)$ and the angle between the line $L$ and the positive direction of the $X$-axis is $\frac{\pi}{3}$,then $p^2 : q^2 =$

Similar Questions

If the direction cosines of two lines satisfy the equations $2l+m-n=0$ and $l^2-2m^2+n^2=0$,and $\theta$ is the angle between the lines,then $\cos \theta=$

If $l, m, n$ are the direction cosines of a line which makes angles $\alpha, \beta$ and $\gamma$ with the coordinate axes $X, Y, Z$,respectively,then $l m+m n+n l$ takes the maximum value when

If $(2, 3, c)$ are the direction ratios of a ray passing through the point $C(5, q, 1)$ and also the midpoint of the line segment joining the points $A(p, -4, 2)$ and $B(3, 2, -4)$,then $c \cdot (p + 7q) = $

If a line makes angles $\frac{\pi}{3}$ and $\frac{\pi}{4}$ with the $X$-axis and $Y$-axis respectively,then the angle made by the line with the $Z$-axis is

If a line lies in the octant $OXYZ$ and it makes equal angles with the axes,then

Vedclass Test Series

Exam Paper Generator

Online Exam Module