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प्रश्न
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)
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उत्तर
Equation of plane through three points (2, 2, 1), (3, 0, 1) and (4, –1, 0) is
`[(vecr - (2hati + 2hatj + hatk)]*[(hati - 2hatj) xx (hati - hatj - hatk)]` = 0
i.e. `vecr*(2hati + hatj + hatk)` = 7 or 2x + y + z – 7 = 0 ......(1)
Equation of line through (3, – 4, – 5) and (2, – 3, 1) is
`(x - 3)/(-1) = (y + 4)/1 = (z + 5)/6` ......(2)
Any point on line (2) is `(-lambda + 3, lambda - 4, 6lambda - 5)`.
This point lies on plane (1).
Therefore, `2(-lambda + 3) + (lambda - 4) + (6lambda - 5) - 7` = 0
i.e., `lambda` = z
Hence the required point is (1, – 2, 7).
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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).
Coordinate planes divide the space into ______ octants.
Name the octants in which the following points lie:
(–5, 4, 3)
Name the octants in which the following points lie:
(4, –3, 5)
Name the octants in which the following points lie:
(–5, –4, 7)
Name the octants in which the following points lie:
(–7, 2 – 5)
Find the image of:
(–2, 3, 4) in the yz-plane.
Find the image of:
(–4, 0, 0) in the xy-plane.
Determine the points in zx-plane are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).
Find the points on z-axis which are at a distance \[\sqrt{21}\]from the point (1, 2, 3).
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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.
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Verify the following:
(–1, 2, 1), (1, –2, 5), (4, –7, 8) and (2, –3, 4) are vertices of a parallelogram.
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.
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Find the ratio in which the line segment joining the points (2, 4,5) and (3, −5, 4) is divided by the yz-plane.
Find the point on x-axis which is equidistant from the points A (3, 2, 2) and B (5, 5, 4).
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 coordinates of the foot of the perpendicular drawn from the point P(3, 4, 5) on the yz- plane are
The coordinates of the foot of the perpendicular from a point P(6,7, 8) on x - axis are
The length of the perpendicular drawn from the point P (3, 4, 5) on y-axis is
The length of the perpendicular drawn from the point P(a, b, c) from z-axis is
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
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 an angle of `pi/4` with each of y and z axis, then the angle which it makes with x-axis is ______.
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.
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 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.
The vector equation of the line `(x - 5)/3 = (y + 4)/7 = (z - 6)/2` is `vecr = (5hati - 4hatj + 6hatk) + lambda(3hati + 7hatj - 2hatk)`.
