Advertisements
Advertisements
प्रश्न
Find the length and the foot of perpendicular from the point `(1, 3/2, 2)` to the plane 2x – 2y + 4z + 5 = 0.
Advertisements
उत्तर
Given plane is 2x – 2y + 4z + 5 = 0 and point `(1, 3/2, 2)`
The direction ratios of the normal to the plane are 2, –2, 4
So, the equation of the line passing through `(1, 3/2, 2)` and direction ratios are equal to the direction ratios of the normal to the plane i.e. 2, –2, 4 is
`(x - 1)/2 = (y - 3/2)/(-2) = (z - 2)/4 = lambda`
Now, any point in the plane is 2λ + 1, –2λ + `3/2`, 4λ + 2
Since, the point lies in the plane, then
2(2λ + 1) – 2(–2λ + `3/2`) + 4(4λ + 2) + 5 = 0
4λ + 2 + 4λ – 3 + 16λ + 8 + 5 = 0
24λ + 12 = 0λ = `1/2`
So, the coordinates of the point in the plane are
`2(-1/2) + 1, -2(-1/2) + 3/2, 4(-1/2) + 2 = 0, 5/2, 0`
Thus, the foot of the perpendicular is (0, 5/2, 0) and the required length
= `sqrt((1 - 0)^2 + (3/2 - 5/2)^2 + (2 - 0)^2)`
= `sqrt(1 + 1 + 4)`
= `sqrt(6)` units
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).
Three vertices of a parallelogram ABCD are A (3, –1, 2), B (1, 2, –4) and C (–1, 1, 2). Find the coordinates of the fourth vertex.
If the origin is the centroid of the triangle PQR with vertices P (2a, 2, 6), Q (–4, 3b, –10) and R (8, 14, 2c), then find the values of a, b and c.
Name the octants in which the following points lie:
(7, 4, –3)
Name the octants in which the following points lie:
(–5, –3, –2)
Name the octants in which the following points lie:
(–7, 2 – 5)
Find the image of:
(–5, 0, 3) in the xz-plane.
Planes are drawn parallel to the coordinate planes through the points (3, 0, –1) and (–2, 5, 4). Find the lengths of the edges of the parallelepiped so formed.
Prove that the triangle formed by joining the three points whose coordinates are (1, 2, 3), (2, 3, 1) and (3, 1, 2) is an equilateral triangle.
Show that the points A(3, 3, 3), B(0, 6, 3), C(1, 7, 7) and D(4, 4, 7) are the vertices of a square.
Prove that the point A(1, 3, 0), B(–5, 5, 2), C(–9, –1, 2) and D(–3, –3, 0) taken in order are the vertices of a parallelogram. Also, show that ABCD is not a rectangle.
Find the coordinates of the point which is equidistant from the four points O(0, 0, 0), A(2, 0, 0), B(0, 3, 0) and C(0, 0, 8).
Verify the following:
(0, 7, –10), (1, 6, –6) and (4, 9, –6) are vertices of an isosceles triangle.
Verify the following:
(0, 7, –10), (1, 6, –6) and (4, 9, –6) are vertices of an isosceles triangle.
Verify the following:
(5, –1, 1), (7, –4,7), (1, –6,10) and (–1, – 3,4) are the vertices of a rhombus.
Show that the points A(1, 2, 3), B(–1, –2, –1), C(2, 3, 2) and D(4, 7, 6) are the vertices of a parallelogram ABCD, but not a rectangle.
Find the ratio in which the sphere x2 + y2 + z2 = 504 divides the line joining the points (12, –4, 8) and (27, –9, 18).
Show that the plane ax + by + cz + d = 0 divides the line joining the points (x1, y1, z1) and (x2, y2, z2) in the ratio \[- \frac{a x_1 + b y_1 + c z_1 + d}{a x_2 + b y_2 + c z_2 + d}\]
Write the distance of the point P(3, 4, 5) from z-axis.
The coordinates of the mid-points of sides AB, BC and CA of △ABC are D(1, 2, −3), E(3, 0,1) and F(−1, 1, −4) respectively. Write the coordinates of its centroid.
The ratio in which the line joining (2, 4, 5) and (3, 5, –9) is divided by the yz-plane is
The ratio in which the line joining the points (a, b, c) and (–a, –c, –b) is divided by the xy-plane is
XOZ-plane divides the join of (2, 3, 1) and (6, 7, 1) in the ratio
What is the locus of a point for which y = 0, z = 0?
The perpendicular distance of the point P(3, 3,4) from the x-axis is
Find the direction cosines of the line passing through the points P(2, 3, 5) and Q(–1, 2, 4).
Find the coordinates of the point where the line through (3, – 4, – 5) and (2, –3, 1) crosses the plane passing through three points (2, 2, 1), (3, 0, 1) and (4, –1, 0)
Prove that the lines x = py + q, z = ry + s and x = p′y + q′, z = r′y + s′ are perpendicular if pp′ + rr′ + 1 = 0.
If the line drawn from the point (–2, – 1, – 3) meets a plane at right angle at the point (1, – 3, 3), find the equation of the plane
O is the origin and A is (a, b, c). Find the direction cosines of the line OA and the equation of plane through A at right angle to OA.
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.
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 locus represented by xy + yz = 0 is ______.
The cartesian equation of the plane `vecr * (hati + hatj - hatk)` is ______.
The unit vector normal to the plane x + 2y +3z – 6 = 0 is `1/sqrt(14)hati + 2/sqrt(14)hatj + 3/sqrt(14)hatk`.
