Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
Given planes are: ax + by = 0 ......(i) and z = 0 ......(ii)
Now, the equation of any plan passing through the line of intersection of plane (i) and (ii) is
(ax + by) + kz = 0
⇒ ax + by + kz = 0 ......(iii)
Dividing both sides by `sqrt(a^2 + b^2 + k^2)`, we get
`a/sqrt(a^2 + b^2 + k^2)x + b/sqrt(a^2 + b&2 + k^2)y + k/sqrt(a^2 + b^2 + k^2)z` = 0
So, direction consines of the normal to the plane are
`a/sqrt(a^2 + b^2 + k^2), b/sqrt(a^2 + b^2 + k^2), k/sqrt(a^2 + b^2 + k^2)`
And the direction cosines of the plane (i) are
`a/sqrt(a^2 + b^2), b/sqrt(a^2 + b^2), 0`
As, α is the angle between the planes (i) and (iii), we get
⇒ cos α = `(a.a + b.b k.0)/(sqrt(a^2 + b^2 + k^2)* sqrt(a^2 + b^2))`
cos α = `(a^2 + b^2)/(sqrt(a^2 + b^2 + k^2) * sqrt(a^2 + b^2))`
cos α = `(a^2 + b^2)/sqrt(a^2 + b^2 + k^2)`
cos2α = `(a^2 + b^2)/sqrt(a^2 + b^2 + k^2)`
`(a^2 + b^2 + k^2) cos^2alpha = a^2 + b^2`
`a^2 cos^2 alpha + b^2 cos^2alpha + k^2 cos^2 alpha = a^2 + b^2`
`k^2 cos^2 alpha = a^2 - a^2 cos^2 alpha 6 b^2 - b^2 cos^2alpha`
`k^2 cos^2 alpha = alpha^2(1 - cos^2alpha)(1 - cos^2alpha)`
`k^2cos^2alpha = a^2 sin^2alpha + b^2 sin^2alpha`
`k^2 cos^2alpha = (a^2 + b^2) sin^2alpha`
⇒ = `k^2 = (a^2 + b^2) (sin^2alpha)/(cos^2alpha)`
⇒ `k = +- sqrt(a^2 + b^2) * tan alpha`
Putting the value of k in equation (iii) we get
`ax + by +- (sqrt(a^2 + b^2) * tan alpha)z` = 0 which is the required equation of plane.
Hence proved.
APPEARS IN
संबंधित प्रश्न
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: (5, 2, 3)
Find the image of:
(–2, 3, 4) in the yz-plane.
Find the image of:
(5, 2, –7) in the xy-plane.
Find the image of:
(–5, 0, 3) in the xz-plane.
Planes are drawn through the points (5, 0, 2) and (3, –2, 5) parallel to the coordinate planes. Find the lengths of the edges of the rectangular parallelepiped so formed.
Find the point on y-axis which is equidistant from the points (3, 1, 2) and (5, 5, 2).
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 locus of P if PA2 + PB2 = 2k2, where A and B are the points (3, 4, 5) and (–1, 3, –7).
Find the locus of the point, the sum of whose distances from the points A(4, 0, 0) and B(–4, 0, 0) is equal to 10.
Find the equation of the set of the points P such that its distances from the points A(3, 4, –5) and B(–2, 1, 4) are equal.
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 \[- \frac{a x_1 + b y_1 + c z_1 + d}{a x_2 + b y_2 + c z_2 + d}\]
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.
Write the coordinates of the foot of the perpendicular from the point (1, 2, 3) on y-axis.
Write the length of the perpendicular drawn from the point P(3, 5, 12) on x-axis.
What is the locus of a point for which y = 0, z = 0?
Find the point on y-axis which is at a distance of \[\sqrt{10}\] units from the point (1, 2, 3).
What is the locus of a point for which y = 0, z = 0?
The coordinates of the foot of the perpendicular drawn from the point P(3, 4, 5) on the yz- plane are
The perpendicular distance of the point P (6, 7, 8) from xy - plane is
The x-coordinate of a point on the line joining the points Q(2, 2, 1) and R(5, 1, –2) is 4. Find its z-coordinate.
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)
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
The coordinates of the foot of the perpendicular drawn from the point (2, 5, 7) on the x-axis are given by ______.
If a line makes angles α, β, γ with the positive directions of the coordinate axes, then the value of sin2α + sin2β + sin2γ is ______.
Find the equation of a plane which bisects perpendicularly the line joining the points A(2, 3, 4) and B(4, 5, 8) at right angles.
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.
If l1, m1, n1 ; l2, m2, n2 ; l3, m3, n3 are the direction cosines of three mutually perpendicular lines, prove that the line whose direction cosines are proportional to l1 + l2 + l3, m1 + m2 + m3, n1 + n2 + n3 makes equal angles with them.
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 locus represented by xy + yz = 0 is ______.
The vector equation of the line through the points (3, 4, –7) and (1, –1, 6) is ______.
The intercepts made by the plane 2x – 3y + 5z +4 = 0 on the co-ordinate axis are `-2, 4/3, - 4/5`.
If the foot of perpendicular drawn from the origin to a plane is (5, – 3, – 2), then the equation of plane is `vecr.(5hati - 3hatj - 2hatk)` = 38.
