Advertisements
Advertisements
प्रश्न
Find the ratio in which the sphere x2 + y2 + z2 = 504 divides the line joining the points (12, –4, 8) and (27, –9, 18).
Advertisements
उत्तर
Let the sphere `x^2+y^2+z^2=504` meet the line joining the given points at the point (x1, y1, z1).
Then, we have:
`x_1^2+y_1^2+z_1^2=504` .............(1)
Suppose the point (x1, y1, z1) divides the line joining the points (12, – 4, 8) and (27, – 9, 18) in the ratio λ:1.
∴`x_1=(27 λ+12)/( λ+1), y_1=(-9λ-4)/(λ+1),z_1=(18λ+8)/(λ+1)`
Substitute these values in equation (1):
`(27λ+12)^2/(λ+1)^2+(-9λ-4)^2/(λ+1)^2+(18λ+8)^2/(λ+1)^2=504`
⇒` 9(9λ+4)^2+(9λ+4)^2+4(9λ+4)^2=504(λ+1)^2`
⇒`9λ+4=+-6(λ+1)`
⇒`λ=+-2/3`
Thus, the sphere divides the line joining the given points in the ratio 2:3 and 2:– 3.
Hence, the given sphere `x^2+y^2+z^2=504` divides the line joining the points (12, – 4, 8) and (27, – 9, 18) internally in the ratio 2:3 and externally in the ratio −2:3.
APPEARS IN
संबंधित प्रश्न
Coordinate planes divide the space into ______ octants.
Three vertices of a parallelogram ABCD are A (3, –1, 2), B (1, 2, –4) and C (–1, 1, 2). Find the coordinates of the fourth vertex.
Name the octants in which the following points lie: (5, 2, 3)
Name the octants in which the following points lie:
(–5, –4, 7)
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:
(–4, 0, 0) 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.
Planes are drawn parallel to the coordinate planes through the points (3, 0, –1) and (–2, 5, 4). Find the lengths of the edges of the parallelepiped so formed.
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.
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 locus of P if PA2 + PB2 = 2k2, where A and B are the points (3, 4, 5) and (–1, 3, –7).
Verify the following:
(0, 7, –10), (1, 6, –6) and (4, 9, –6) are vertices of an isosceles triangle.
Verify the following:
(5, –1, 1), (7, –4,7), (1, –6,10) and (–1, – 3,4) are the vertices of a rhombus.
Show that the points A(1, 2, 3), B(–1, –2, –1), C(2, 3, 2) and D(4, 7, 6) are the vertices of a parallelogram ABCD, but not a rectangle.
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.
Find the point on y-axis which is at a distance of \[\sqrt{10}\] units from the point (1, 2, 3).
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 (2, 4, 5) and (3, 5, –9) is divided by the yz-plane is
The ratio in which the line joining the points (a, b, c) and (–a, –c, –b) is divided by the xy-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
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 ______.
If a line makes angles α, β, γ with the positive directions of the coordinate axes, then the value of sin2α + sin2β + sin2γ 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.
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 foot of perpendicular from the point (2,3,–8) to the line `(4 - x)/2 = y/6 = (1 - z)/3`. Also, find the perpendicular distance from the given point to the line.
Find the equation of the plane through the points (2, 1, –1) and (–1, 3, 4), and perpendicular to the plane x – 2y + 4z = 10.
The plane ax + by = 0 is rotated about its line of intersection with the plane z = 0 through an angle α. Prove that the equation of the plane in its new position is ax + by `+- (sqrt(a^2 + b^2) tan alpha)z ` = 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.
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 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 vector equation of the line `(x - 5)/3 = (y + 4)/7 = (z - 6)/2` is `vecr = (5hati - 4hatj + 6hatk) + lambda(3hati + 7hatj - 2hatk)`.
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.
