Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Let P (x, y, z) be any point , the sum of whose distance from the points A(4,0,0) and B(\[-\]4,0,0)
is equal to 10. Then, PA + PB = 10
\[\Rightarrow \sqrt{\left( x - 4 \right)^2 + \left( y - 0 \right)^2 + \left( z - 0 \right)^2} + \sqrt{\left( x + 4 \right)^2 + \left( y - 0 \right)^2 + \left( z - 0 \right)^2} = 10\]
\[ \Rightarrow \sqrt{\left( x + 4 \right)^2 + \left( y - 0 \right)^2 + \left( z - 0 \right)^2} = 10 - \sqrt{\left( x - 4 \right)^2 + \left( y - 0 \right)^2 + \left( z - 0 \right)^2}\]
\[ \Rightarrow \sqrt{x^2 + 8x + 16 + y^2 + z^2} = 10 - \sqrt{x^2 - 8x + 16 + y^2 + z^2}\]
\[\Rightarrow x^2 + 8x + 16 + y^2 + z^2 = 100 + x^2 - 8x + 16 + y^2 + z^2 - 20\sqrt{x^2 - 8x + 16 + y^2 + z^2}\]
\[ \Rightarrow 16x - 100 = - 20\sqrt{x^2 - 8x + 16 + y^2 + z^2}\]
\[ \Rightarrow 4x - 25 = - 5\sqrt{x^2 - 8x + 16 + y^2 + z^2}\]
\[ \Rightarrow 16 x^2 - 200x + 625 = 25\left( x^2 - 8x + 16 + y^2 + z^2 \right)\]
\[ \Rightarrow 16 x^2 - 200x + 625 = 25 x^2 - 200x + 400 + 25 y^2 + 25 z^2 \]
\[ \Rightarrow 9 x^2 + 25 y^2 + 25 z^2 - 225 = 0\]
\[\therefore 9 x^2 + 25 y^2 + 25 z^2 - 225 = 0 \text{ is the required locus } .\]
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:
(7, 4, –3)
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:
(–5, 4, –3) in the xz-plane.
Find the image of:
(–5, 0, 3) in the xz-plane.
Find the image of:
(–4, 0, 0) in the xy-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 points in zx-plane are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).
Determine the point on z-axis which is equidistant from the points (1, 5, 7) and (5, 1, –4).
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.
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.
Show that the points (a, b, c), (b, c, a) and (c, a, b) are the vertices of an equilateral triangle.
Find the locus of the points which are equidistant from the points (1, 2, 3) and (3, 2, –1).
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 point on y-axis which is at a distance of \[\sqrt{10}\] units from the point (1, 2, 3).
The ratio in which the line joining the points (a, b, c) and (–a, –c, –b) is divided by the xy-plane is
The coordinates of the foot of the perpendicular from a point P(6,7, 8) on x - axis are
If the direction ratios of a line are 1, 1, 2, find the direction cosines of the line.
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 a line makes angles `pi/2, 3/4 pi` and `pi/4` with x, y, z axis, respectively, then its direction cosines 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 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.
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.
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 locus represented by xy + yz = 0 is ______.
