Advertisements
Advertisements
प्रश्न
Ratio in which the xy-plane divides the join of (1, 2, 3) and (4, 2, 1) is
पर्याय
3 : 1 internally
3 : 1 externally
1 : 2 internally
2 : 1 externally
Advertisements
उत्तर
` 3: 1 \text{ externally } `
\[\text{ Suppose the XY - plane divides the line segment joining the points P } \left( 1, 2, 3 \right) \text{ and Q } \left( 4, 2, 1 \right) \text{ in the ratio k: 1 } . \]
\[\text{ Using the section formula, the coordinates of the point of intersection are given by } \]
\[\left( \frac{k\left( 4 \right) + 1}{k + 1}, \frac{k\left( 2 \right) + 2}{k + 1}, \frac{k\left( 1 \right) + 3}{k + 1} \right)\]
\[\text{ The Z - coordinate of any point on the XY - plane is zero }. \]
\[ \Rightarrow \frac{k\left( 1 \right) + 3}{k + 1} = 0\]
\[ \Rightarrow k + 3 = 0\]
\[ \Rightarrow k = - 3 = - \frac{3}{1}\]
\[\text{ Thus, the XY - plane divides the line segment joining the given points in the ratio 3: 1 externally } . \]
APPEARS IN
संबंधित प्रश्न
Find the direction cosines of the line
`(x+2)/2=(2y-5)/3; z=-1`
Write the direction ratios of the following line :
`x = −3, (y−4)/3 =( 2 −z)/1`
If l, m, n are the direction cosines of a line, then prove that l2 + m2 + n2 = 1. Hence find the
direction angle of the line with the X axis which makes direction angles of 135° and 45° with Y and Z axes respectively.
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.
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).
If the coordinates of the points A, B, C, D are (1, 2, 3), (4, 5, 7), (−4, 3, −6) and (2, 9, 2), then find the angle between AB and CD.
What are the direction cosines of Y-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 ratio in which the line segment joining (a, b, c) and (−a, −c, −b) is divided by the xy-plane.
For every point P (x, y, z) on the xy-plane,
For every point P (x, y, z) on the x-axis (except the origin),
Find the equation of the lines passing through the point (2, 1, 3) and perpendicular to the lines
If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines.
Verify whether the following ratios are direction cosines of some vector or not
`1/5, 3/5, 4/5`
Find the direction cosines and direction ratios for the following vector
`3hat"i" + hat"j" + hat"k"`
If `1/2, 1/sqrt(2), "a"` are the direction cosines of some vector, then find a
Choose the correct alternative:
The unit vector parallel to the resultant of the vectors `hat"i" + hat"j" - hat"k"` and `hat"i" - 2hat"j" + hat"k"` is
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.
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 ______.
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 ______.
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.
The area of the quadrilateral ABCD, where A(0,4,1), B(2, 3, –1), C(4, 5, 0) and D(2, 6, 2), is equal to ______.
The direction cosines of vector `(2hat"i" + 2hat"j" - hat"k")` are ______.
The line `vec"r" = 2hat"i" - 3hat"j" - hat"k" + lambda(hat"i" - hat"j" + 2hat"k")` lies in the plane `vec"r".(3hat"i" + hat"j" - hat"k") + 2` = 0.
The Cartesian equation of a line AB is: `(2x - 1)/2 = (y + 2)/2 = (z - 3)/3`. Find the direction cosines of a line parallel to line AB.
The co-ordinates of the point where the line joining the points (2, –3, 1), (3, –4, –5) cuts the plane 2x + y + z = 7 are ______.
Equation of a line passing through point (1, 2, 3) and equally inclined to the coordinate axis, is ______.
If a line makes angles of 90°, 135° and 45° with the x, y and z axes respectively, then its direction cosines are ______.
