Advertisements
Advertisements
प्रश्न
Find the equations of the two lines through the origin which intersect the line `(x - 3)/2 = (y - 3)/1 = z/1` at angles of `pi/3` each.
Advertisements
उत्तर
Any point on the given line is `(x - 3)/2 = (y - 3)/1 = z/1 = lambda`
⇒ x = 2λ + 3, y = λ + 3
And z = λ
Let it be the coordinates of P
∴ Direction ratios of OP are (2λ + 3 – 0), (λ + 3 – 0) and (λ – 0)
⇒ 2λ + 3, λ + 3, λ
But the direction ratios of the line PQ are 2, 1, 1
∴ `cos theta = ("a"_1"a"_2 + "b"_1"b"_2 + "c"_1"c"_2)/(sqrt("a"_1^2 + "b"_1^2 + "c"_1^2)*sqrt("a"_2^2 + "b"_2^2 + "c"_2^2)`
`cos pi/3 = (2(2lambda + 3) + 1(lambda + 3) + 1.lambda)/(sqrt((2)^2 + (1)^2 + (1)^2) * sqrt((2lambda + 3)^2 + (lambda + 3)^2 + lambda^2)`
⇒ `1/2 = (4lambda + 6 + lambda + 3 + lambda)/(sqrt(6) * sqrt(4lambda^2 + 9 + 12lambda + lambda^2 + 9 + 6lambda + lambda^2)`
⇒ `sqrt(6)/2 = (6lambda + 9)/sqrt(6lambda^2 + 18lambda + 18)`
= `(6lambda + 9)/(sqrt(6)sqrt(lambda^2 + 3lambda + 3)`
⇒ `6/2 = (3(2lambda + 3))/sqrt(lambda^2 + 3lambda + 3)`
⇒ 3 = `(3(2lambda + 3))/sqrt(lambda^2 + 3lambda + 3)`
⇒ 1 = `(2lambda + 3)/sqrt(lambda^2 + 3lambda + 3)`
⇒ `sqrt(lambda^2 + 3lambda + 3) = 2lambda + 3`
⇒ λ2 + 3λ+ 3 = 4λ2 + 9 + 12λ ......(Squaring both sides)
⇒ 3λ2 + 9λ + 6 = 0
⇒ λ2 + 3λ + 2 = 0
⇒ (λ + 1)(λ + 2) = 0
∴ λ = – 1, λ = – 2
∴ Direction ratios are [2(– 1) + 3, – 1 + 3, – 1]
i.e., 1, 2, – 1
When λ = – 1 and [2(– 2) + 3, – 2 + 3, – 2]
i.e., – 1, 1, – 2
When λ = – 2.
Hence, the required equations are
`x/1 = y/2 = z/(-1)` and `x/(-1) = y/1 = z/(-2)`.
APPEARS IN
संबंधित प्रश्न
Find the direction cosines of the line perpendicular to the lines whose direction ratios are -2, 1,-1 and -3, - 4, 1
Direction cosines of the line passing through the points A (- 4, 2, 3) and B (1, 3, -2) are.........
Which of the following represents direction cosines of the line :
(a)`0,1/sqrt2,1/2`
(b)`0,-sqrt3/2,1/sqrt2`
(c)`0,sqrt3/2,1/2`
(d)`1/2,1/2,1/2`
Find the angle between the lines whose direction ratios are 4, –3, 5 and 3, 4, 5.
If a line has the direction ratios −18, 12, −4, then what are its direction cosines?
Find the vector equation of the plane passing through (1, 2, 3) and perpendicular to the plane `vecr.(hati + 2hatj -5hatk) + 9 = 0`
If a line has direction ratios 2, −1, −2, determine its direction cosines.
Find the direction cosines of the sides of the triangle whose vertices are (3, 5, −4), (−1, 1, 2) and (−5, −5, −2).
Find the acute angle between the lines whose direction ratios are proportional to 2 : 3 : 6 and 1 : 2 : 2.
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
Show that the line through points (4, 7, 8) and (2, 3, 4) is parallel to the line through the points (−1, −2, 1) and (1, 2, 5).
Show that the line through the points (1, −1, 2) and (3, 4, −2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).
Find the angle between the lines whose direction ratios are proportional to a, b, c and b − c, c − a, a− b.
Find the angle between the lines whose direction cosines are given by the equations
(i) l + m + n = 0 and l2 + m2 − n2 = 0
Write the distances of the point (7, −2, 3) from XY, YZ and XZ-planes.
Write the distance of the point (3, −5, 12) from X-axis?
Write the ratio in which YZ-plane divides the segment joining P (−2, 5, 9) and Q (3, −2, 4).
If a line makes angles α, β and γ with the coordinate axes, find the value of cos2α + cos2β + cos2γ.
Write the inclination of a line with Z-axis, if its direction ratios are proportional to 0, 1, −1.
Write the coordinates of the projection of point P (x, y, z) on XOZ-plane.
For every point P (x, y, z) on the xy-plane,
The xy-plane divides the line joining the points (−1, 3, 4) and (2, −5, 6)
If the x-coordinate of a point P on the join of Q (2, 2, 1) and R (5, 1, −2) is 4, then its z-coordinate is
If a line makes angles α, β, γ, δ with four diagonals of a cube, then cos2 α + cos2 β + cos2γ + cos2 δ is equal to
Verify whether the following ratios are direction cosines of some vector or not
`1/5, 3/5, 4/5`
Verify whether the following ratios are direction cosines of some vector or not
`1/sqrt(2), 1/2, 1/2`
Find the direction cosines of a vector whose direction ratios are
0, 0, 7
If `1/2, 1/sqrt(2), "a"` are the direction cosines of some vector, then find a
If (a, a + b, a + b + c) is one set of direction ratios of the line joining (1, 0, 0) and (0, 1, 0), then find a set of values of a, b, c
P is a point on the line segment joining the points (3, 2, –1) and (6, 2, –2). If x co-ordinate of P is 5, then its y co-ordinate is ______.
The vector equation of the line passing through the points (3, 5, 4) and (5, 8, 11) is `vec"r" = 3hat"i" + 5hat"j" + 4hat"k" + lambda(2hat"i" + 3hat"j" + 7hat"k")`
If the directions cosines of a line are k,k,k, then ______.
If a line has the direction ratio – 18, 12, – 4, then what are its direction cosine.
If a line makes an angle α, β and γ with positive direction of the coordinate axes, then the value of sin2α + sin2β + sin2γ will be ______.
