Advertisements
Advertisements
प्रश्न
The xy-plane divides the line joining the points (−1, 3, 4) and (2, −5, 6)
पर्याय
internally in the ratio 2 : 3
externally in the ratio 2 : 3
internally in the ratio 3 : 2
externally in the ratio 3 : 2
Advertisements
उत्तर
`\text{ externally in the ratio 2: 3 } `
\[\text{ Let the XY - plane divide the line segment joining points }P\left( - 1, 3, 4 \right) \text{ and } Q\left( 2, - 5, 6 \right) \text{ in the ratio k: 1 }. \]
\[\text { Using the section formula, the coordinates of the point of intersection are given by } \]
\[\left( \frac{k\left( 2 \right) - 1}{k + 1}, \frac{k\left( - 5 \right) + 3}{k + 1}, \frac{k\left( 6 \right) + 4}{k + 1} \right)\]
\[\text { On the XY - plane, the Z - coordinate of any point is zero } . \]
\[ \Rightarrow \frac{k\left( 6 \right) + 4}{k + 1} = 0\]
\[ \Rightarrow 6k + 4 = 0\]
\[ \Rightarrow k = - \frac{2}{3}\]
\[\text{ Thus, the XY - plane divides the line segment joining the given points in the ratio 2: 3 externally } . \]
APPEARS IN
संबंधित प्रश्न
Find the direction cosines of the line
`(x+2)/2=(2y-5)/3; z=-1`
Direction cosines of the line passing through the points A (- 4, 2, 3) and B (1, 3, -2) are.........
Which of the following represents direction cosines of the line :
(a)`0,1/sqrt2,1/2`
(b)`0,-sqrt3/2,1/sqrt2`
(c)`0,sqrt3/2,1/2`
(d)`1/2,1/2,1/2`
If a line makes angles of 90°, 60° and 30° with the positive direction of x, y, and z-axis respectively, find its direction cosines
Using direction ratios show that the points A (2, 3, −4), B (1, −2, 3) and C (3, 8, −11) are collinear.
Find the acute angle between the lines whose direction ratios are proportional to 2 : 3 : 6 and 1 : 2 : 2.
Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, −1) and (4, 3, −1).
Find the direction cosines of the lines, connected by the relations: l + m +n = 0 and 2lm + 2ln − mn= 0.
Find the angle between the lines whose direction cosines are given by the equations
l + 2m + 3n = 0 and 3lm − 4ln + mn = 0
Find the angle between the lines whose direction cosines are given by the equations
2l + 2m − n = 0, mn + ln + lm = 0
What are the direction cosines of X-axis?
Write the ratio in which YZ-plane divides the segment joining P (−2, 5, 9) and Q (3, −2, 4).
Write the coordinates of the projection of point P (x, y, z) on XOZ-plane.
Write the coordinates of the projection of the point P (2, −3, 5) on Y-axis.
If a line has direction ratios proportional to 2, −1, −2, then what are its direction consines?
Write direction cosines of a line parallel to z-axis.
Ratio in which the xy-plane divides the join of (1, 2, 3) and (4, 2, 1) is
If a line makes angles α, β, γ, δ with four diagonals of a cube, then cos2 α + cos2 β + cos2γ + cos2 δ is equal to
Verify whether the following ratios are direction cosines of some vector or not
`1/5, 3/5, 4/5`
Find the direction cosines of a vector whose direction ratios are
`1/sqrt(2), 1/2, 1/2`
Find the direction cosines and direction ratios for the following vector
`3hat"i" + hat"j" + hat"k"`
Find the direction cosines and direction ratios for the following vector
`5hat"i" - 3hat"j" - 48hat"k"`
Find the direction cosines and direction ratios for the following vector
`hat"i" - hat"k"`
A triangle is formed by joining the points (1, 0, 0), (0, 1, 0) and (0, 0, 1). Find the direction cosines of the medians
If (a, a + b, a + b + c) is one set of direction ratios of the line joining (1, 0, 0) and (0, 1, 0), then find a set of values of a, b, c
Find the direction cosines of the line passing through the points P(2, 3, 5) and Q(–1, 2, 4).
A line makes equal angles with co-ordinate axis. Direction cosines of this line are ______.
Find the equations of the two lines through the origin which intersect the line `(x - 3)/2 = (y - 3)/1 = z/1` at angles of `pi/3` each.
If a variable line in two adjacent positions has direction cosines l, m, n and l + δl, m + δm, n + δn, show that the small angle δθ between the two positions is given by δθ2 = δl2 + δm2 + δn2
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.
What will be the value of 'P' so that the lines `(1 - x)/3 = (7y - 14)/(2P) = (z - 3)/2` and `(7 - 7x)/(3P) = (y - 5)/1 = (6 - z)/5` at right angles.
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.
A line passes through the points (6, –7, –1) and (2, –3, 1). The direction cosines of the line so directed that the angle made by it with positive direction of x-axis is acute, are ______.
The projections of a vector on the three coordinate axis are 6, –3, 2 respectively. The direction cosines of the vector are ______.
If the equation of a line is x = ay + b, z = cy + d, then find the direction ratios of the line and a point on the line.
Find the coordinates of the foot of the perpendicular drawn from point (5, 7, 3) to the line `(x - 15)/3 = (y - 29)/8 = (z - 5)/-5`.
Find the coordinates of the image of the point (1, 6, 3) with respect to the line `vecr = (hatj + 2hatk) + λ(hati + 2hatj + 3hatk)`; where 'λ' is a scalar. Also, find the distance of the image from the y – axis.
