Advertisements
Advertisements
प्रश्न
Find the foot of perpendicular from the point (2,3,–8) to the line `(4 - x)/2 = y/6 = (1 - z)/3`. Also, find the perpendicular distance from the given point to the line.
उत्तर
Let the given equation be `(4 - x)/2 = y/6 = (1 - z)/3 = lambda`
This can be written as `(x - 4)/(-2) = y/6 = (z - 1)/(-3) = lambda` .......(1)
∴ The coordinates of any point on the line is x = 4 − 2λ, y = 6λ, z = 1 − 3λ
Let Q(4 − 2λ, 6λ, 1 − 3λ) be the foot of perpendicular from the point P(2, 3, −8) on line ......(1)
We know the direction ratios of any line segement PQ is given by (x2 − x1, y2 − y1, z2 − z1)
The direction cosines of PQ is given by
= (−2λ + 4 − 2, 6λ − 3, −3λ + 1 + 8)
= (−2λ + 2, 6λ − 3, −3λ + 9)
Now Q is the foot of the perpendicular of the line (1)
`vec(PQ)` is the perpendicular to the line (1)
Hence the sum of the product of this direction ratios is 0
= (−2λ + 2)(−2) + (6λ − 3) . 6 + (−3λ + 9)(−3) = 0
⇒ 4λ − 4 + 36λ − 18 + 9λ − 27 = 0
APPEARS IN
संबंधित प्रश्न
Name the octants in which the following points lie:
(1, 2, 3), (4, –2, 3), (4, –2, –5), (4, 2, –5), (–4, 2, –5), (–4, 2, 5),
(–3, –1, 6), (2, –4, –7).
Coordinate planes divide the space into ______ octants.
Name the octants in which the following points lie: (5, 2, 3)
Name the octants in which the following points lie:
(7, 4, –3)
Name the octants in which the following points lie:
(–5, –4, 7)
Find the image of:
(5, 2, –7) in the xy-plane.
Determine the points in zx-plane are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).
Determine the point on z-axis which is equidistant from the points (1, 5, 7) and (5, 1, –4).
Find the point on y-axis which is equidistant from the points (3, 1, 2) and (5, 5, 2).
Verify the following:
(5, –1, 1), (7, –4,7), (1, –6,10) and (–1, – 3,4) are the vertices of a rhombus.
Find the locus of the points which are equidistant from the points (1, 2, 3) and (3, 2, –1).
Write the distance of the point P (2, 3,5) from the xy-plane.
Write the distance of the point P(3, 4, 5) from z-axis.
XOZ-plane divides the join of (2, 3, 1) and (6, 7, 1) in the ratio
If the direction ratios of a line are 1, 1, 2, find the direction cosines of the line.
Find the direction cosines of the line passing through the points P(2, 3, 5) and Q(–1, 2, 4).
If a line makes an angle of 30°, 60°, 90° with the positive direction of x, y, z-axes, respectively, then find its direction cosines.
A plane meets the co-ordinates axis in A, B, C such that the centroid of the ∆ABC is the point (α, β, γ). Show that the equation of the plane is `x/alpha + y/beta + z/γ` = 3
If α, β, γ are the angles that a line makes with the positive direction of x, y, z axis, respectively, then the direction cosines of the line are ______.
A line makes equal angles with co-ordinate axis. Direction cosines of this line are ______.
If a line makes angles `pi/2, 3/4 pi` and `pi/4` with x, y, z axis, respectively, then its direction cosines are ______.
Find the length and the foot of perpendicular from the point `(1, 3/2, 2)` to the plane 2x – 2y + 4z + 5 = 0.
Find the equations of the line passing through the point (3,0,1) and parallel to the planes x + 2y = 0 and 3y – z = 0.
Find the equation of the plane through the points (2, 1, –1) and (–1, 3, 4), and perpendicular to the plane x – 2y + 4z = 10.
Find the equation of the plane which is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which contains the line of intersection of the planes x + 2y + 3z – 4 = 0 and 2x + y – z + 5 = 0.
The plane ax + by = 0 is rotated about its line of intersection with the plane z = 0 through an angle α. Prove that the equation of the plane in its new position is ax + by `+- (sqrt(a^2 + b^2) tan alpha)z ` = 0
If the directions cosines of a line are k, k, k, then ______.
The sine of the angle between the straight line `(x - 2)/3 = (y - 3)/4 = (z - 4)/5` and the plane 2x – 2y + z = 5 is ______.
The cartesian equation of the plane `vecr * (hati + hatj - hatk)` is ______.
