Advertisements
Advertisements
प्रश्न
Two systems of rectangular axis have the same origin. If a plane cuts them at distances a, b, c and a′, b′, c′, respectively, from the origin, prove that
`1/a^2 + 1/b^2 + 1/c^2 = 1/(a"'"^2) + 1/(b"'"^2) + 1/(c"'"^2)`
उत्तर
Let’s take OX, OY, OZ and ox, oy, oz to be two rectangular systems.
And, the equations of two planes are
`"X"/a + "Y"/b + "Z"/c` = 1 ......(i)
And `x/(a"'") + y/(b"'") + z/(c"'")` = 1 .....(ii)
Length of perpendicular from origin to plane (i) is
= `|(0/a + 0/b + 0/c - 1)/sqrt(1/a^2 + 1/b^2 + 1/c^2)|`
= `1/sqrt(1/a^2 + 1/b^2 + 1/c^2)`
Length of perpendicular from origin to plane (ii)
= `|(0/(a"'") + 0/(b"'") + 0/(c"'") - 1)/sqrt(1/(a"'"^2) + 1/(b"'"^2) + 1/(c"'"^2))|`
= `1/sqrt(1/(a"'"^2) + 1/(b"'"^2) + 1/(c"'"^2))`
As per the condition of the question
`1/sqrt(1/a^2 + 1/b^2 + 1/c^2) = 1/sqrt(1/(a"'"^2) + 1/(b"'"^2) + 1/(c"'"^2))`
Thus, `1/a^2 + 1/b^2 + 1/c^2 = 1/(a"'"^2) + 1/(b"'"^2) + 1/(c"'"^2)`
APPEARS IN
संबंधित प्रश्न
Name the octants in which the following points lie:
(–7, 2 – 5)
Find the image of:
(–2, 3, 4) in the yz-plane.
A cube of side 5 has one vertex at the point (1, 0, –1), and the three edges from this vertex are, respectively, parallel to the negative x and y axes and positive z-axis. Find the coordinates of the other vertices of the cube.
Find the point on y-axis which is equidistant from the points (3, 1, 2) and (5, 5, 2).
Find the points on z-axis which are at a distance \[\sqrt{21}\]from the point (1, 2, 3).
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).
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.
Write the length of the perpendicular drawn from the point P(3, 5, 12) on x-axis.
The ratio in which the line joining (2, 4, 5) and (3, 5, –9) is divided by the yz-plane is
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 perpendicular distance of the point P(3, 3,4) from the x-axis is
If the direction ratios of a line are 1, 1, 2, find the direction cosines of the line.
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.
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 ______.
If a line makes angles `pi/2, 3/4 pi` and `pi/4` with x, y, z axis, respectively, then its direction cosines are ______.
If a line makes an angle of `pi/4` with each of y and z axis, then the angle which it makes with x-axis is ______.
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.
Show that the points `(hati - hatj + 3hatk)` and `3(hati + hatj + hatk)` are equidistant from the plane `vecr * (5hati + 2hatj - 7hatk) + 9` = 0 and lies on opposite side of it.
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 direction cosines of the vector `(2hati + 2hatj - hatk)` are ______.
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`.
The intercepts made by the plane 2x – 3y + 5z +4 = 0 on the co-ordinate axis are `-2, 4/3, - 4/5`.
