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:
(–5, 4, 3)
Name the octants in which the following points lie:
(2, –5, –7)
Find the image of:
(–2, 3, 4) in the yz-plane.
Find the image of:
(5, 2, –7) in the xy-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 distances of the point P(–4, 3, 5) from the coordinate axes.
Find the points on z-axis which are at a distance \[\sqrt{21}\]from the point (1, 2, 3).
Show that the points (a, b, c), (b, c, a) and (c, a, b) are the vertices of an equilateral triangle.
Verify the following:
(0, 7, –10), (1, 6, –6) and (4, 9, –6) are vertices of an isosceles triangle.
Find the locus of the points which are equidistant from the points (1, 2, 3) and (3, 2, –1).
Find the locus of the point, the sum of whose distances from the points A(4, 0, 0) and B(–4, 0, 0) is equal to 10.
Write the distance of the point P (2, 3,5) from the xy-plane.
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.
The ratio in which the line joining (2, 4, 5) and (3, 5, –9) is divided by the yz-plane is
Let (3, 4, –1) and (–1, 2, 3) be the end points of a diameter of a sphere. Then, the radius of the sphere is equal to
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.
Find the co-ordinates of the foot of perpendicular drawn from the point A(1, 8, 4) to the line joining the points B(0, –1, 3) and C(2, –3, –1).
The coordinates of the foot of the perpendicular drawn from the point (2, 5, 7) on the x-axis are given by ______.
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 ______.
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.
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.
If the line drawn from the point (–2, – 1, – 3) meets a plane at right angle at the point (1, – 3, 3), find the equation of the plane
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.
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.
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)`
Find the length and the foot of perpendicular from the point `(1, 3/2, 2)` to the plane 2x – 2y + 4z + 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 l1, m1, n1 ; l2, m2, n2 ; l3, m3, n3 are the direction cosines of three mutually perpendicular lines, prove that the line whose direction cosines are proportional to l1 + l2 + l3, m1 + m2 + m3, n1 + n2 + n3 makes equal angles with them.
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 vector equation of the line through the points (3, 4, –7) and (1, –1, 6) is ______.
The angle between the line `vecr = (5hati - hatj - 4hatk) + lambda(2hati - hatj + hatk)` and the plane `vec.(3hati - 4hatj - hatk)` + 5 = 0 is `sin^-1(5/(2sqrt(91)))`.
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.
