Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Let P (x, y, z) be any point which is equidistant from A (3,4,5) and B (\[-\]2,1,4) .Then,
PA = PB\[\Rightarrow \sqrt{\left( x - 3 \right)^2 + \left( y - 4 \right)^2 + \left( z + 5 \right)^2} = \sqrt{\left( x + 2 \right)^2 + \left( y - 1 \right)^2 + \left( z - 4 \right)^2}\]
\[ \Rightarrow \sqrt{x^2 - 6x + 9 + y^2 - 8y + 16 + z^2 + 10z + 25} = \sqrt{x^2 + 4x + 4 + y^2 - 2y + 1 + z^2 - 8z + 16}\]
\[ \Rightarrow x^2 - 6x + 9 + y^2 - 8y + 16 + z^2 + 10z + 25 = x^2 + 4x + 4 + y^2 - 2y + 1 + z^2 - 8z + 16\]
\[ \Rightarrow - 10x - 6y + 18z + 29 = 0\]
\[ \therefore 10x + 6y - 18z - 29 = 0\]
Hence, 10x + 6y\[-\]18z\[-\]29 = 0 is the required equation.
APPEARS IN
संबंधित प्रश्न
The x-axis and y-axis taken together determine a plane known as_______.
Coordinate planes divide the space into ______ octants.
Name the octants in which the following points lie:
(4, –3, 5)
Name the octants in which the following points lie:
(2, –5, –7)
Find the distances of the point P(–4, 3, 5) from the coordinate axes.
The coordinates of a point are (3, –2, 5). Write down the coordinates of seven points such that the absolute values of their coordinates are the same as those of the coordinates of the given point.
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.
Show that the points (a, b, c), (b, c, a) and (c, a, b) are the vertices of an equilateral triangle.
Verify the following:
(–1, 2, 1), (1, –2, 5), (4, –7, 8) and (2, –3, 4) are vertices of a parallelogram.
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.
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 ratio in which the sphere x2 + y2 + z2 = 504 divides the line joining the points (12, –4, 8) and (27, –9, 18).
Show that the plane ax + by + cz + d = 0 divides the line joining the points (x1, y1, z1) and (x2, y2, z2) in the ratio \[- \frac{a x_1 + b y_1 + c z_1 + d}{a x_2 + b y_2 + c z_2 + d}\]
Write the distance of the point P (2, 3,5) from the xy-plane.
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 y-axis which is at a distance of \[\sqrt{10}\] units from the point (1, 2, 3).
What is the locus of a point for which y = 0, z = 0?
The length of the perpendicular drawn from the point P (3, 4, 5) on y-axis is
The perpendicular distance of the point P(3, 3,4) from the x-axis is
Find the direction cosines of the line passing through the points P(2, 3, 5) and Q(–1, 2, 4).
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 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 the plane through the points (2, 1, 0), (3, –2, –2) and (3, 1, 7).
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.
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.
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.
Show that the straight lines whose direction cosines are given by 2l + 2m – n = 0 and mn + nl + lm = 0 are at right angles.
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 locus represented by xy + yz = 0 is ______.
The direction cosines of the vector `(2hati + 2hatj - hatk)` are ______.
The vector equation of the line `(x - 5)/3 = (y + 4)/7 = (z - 6)/2` is ______.
The line `vecr = 2hati - 3hatj - hatk + lambda(hati - hatj + 2hatk)` lies in the plane `vecr.(3hati + hatj - hatk) + 2` = 0.
