Advertisements
Advertisements
प्रश्न
Find the point on y-axis which is equidistant from the points (3, 1, 2) and (5, 5, 2).
Advertisements
उत्तर
Let the point on the y-axis be Y\[\left( 0, y, 0 \right)\]which is equidistant from the points P\[\left( 3, 1, 2 \right)\]and Q \[\left( 5, 5, 2 \right)\]
Then, PY = QY
Now,
`sqrt((3 - 0)^2 + (1 - y)^2 + (2 - 0)^2) = sqrt((5 - 0)^2 + (5 - y)^2 + (2 - 0)^2)`
`=> sqrt((3)^2 + (1 - y)^2 + (2)^2) = sqrt((5)^2 + (5 - y)^2 + (2)^2)`
`=> sqrt(9 + (1 - y)^2 + 4) = sqrt(25 + (5 - y)^2 + 4)`
`=> 9 + (1 - y)^2 + 4 = 25 + (5 - y)^2 + 4`
`=> 9 + (1 - y)^2 + cancel(4) = 25 + (5 - y)^2 + cancel(4)`
`=> 9 + (1 - y)^2 = 25 + (5 - y)^2`
`=> 1 + y^2 - 2y = 25 - 9 + (5 - y)^2`
`=> 1 + y^2 - 2y = 16 + 25 + y^2 - 10y`
`=> 1 + cancel(y^2) - 2y = 41 + cancel(y^2) - 10y`
`=> - 2y = 41 - 1 - 10y`
`=> - 2y = 40 - 10y`
⇒ 8y = 40
⇒ y = `40/8`
⇒ y = 5
Thus, the required point on the y-axis is (0, 5, 0).
APPEARS IN
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, 2, 3)
Find the image of:
(–5, 4, –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 distances of the point P(–4, 3, 5) from the coordinate axes.
Determine the point on z-axis which is equidistant from the points (1, 5, 7) and (5, 1, –4).
Prove that the triangle formed by joining the three points whose coordinates are (1, 2, 3), (2, 3, 1) and (3, 1, 2) is an equilateral triangle.
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 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).
Verify the following:
(0, 7, 10), (–1, 6, 6) and (–4, 9, –6) are vertices of a right-angled triangle.
Verify the following:
(5, –1, 1), (7, –4,7), (1, –6,10) and (–1, – 3,4) are the vertices of a rhombus.
Find the locus of the points which are equidistant from the points (1, 2, 3) and (3, 2, –1).
Write the distance of the point P (2, 3,5) from the xy-plane.
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 x-axis which is equidistant from the points A (3, 2, 2) and B (5, 5, 4).
The length of the perpendicular drawn from the point P (3, 4, 5) on y-axis is
If the direction ratios of a line are 1, 1, 2, find the direction cosines of the line.
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)
The coordinates of the foot of the perpendicular drawn from the point (2, 5, 7) on the x-axis are given by ______.
A line makes equal angles with co-ordinate axis. Direction cosines of this line are ______.
If a line makes angles α, β, γ with the positive directions of the coordinate axes, then the value of sin2α + sin2β + sin2γ is ______.
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 a plane which bisects perpendicularly the line joining the points A(2, 3, 4) and B(4, 5, 8) at right angles.
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.
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)`
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.
Show that the straight lines whose direction cosines are given by 2l + 2m – n = 0 and mn + nl + lm = 0 are at right angles.
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 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 vector equation of the line through the points (3, 4, –7) and (1, –1, 6) is ______.
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`.
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.
