Advertisements
Advertisements
प्रश्न
If A(–2, 2, 3) and B(13, –3, 13) are two points.
Find the locus of a point P which moves in such a way the 3PA = 2PB.
Advertisements
उत्तर
Let coordinates of point P be (x, y, z).
Given:
3PA = 2PB
\[\Rightarrow 3\left( \sqrt{\left( x + 2 \right)^2 + \left( y - 2 \right)^2 + \left( z - 3 \right)^2} \right) = 2\left( \sqrt{\left( x - 13 \right)^2 + \left( y + 3 \right)^2 + \left( z - 13 \right)^2} \right)\]
\[ \Rightarrow 3\left( \sqrt{x^2 + 4x + 4 + y^2 - 4y + 4 + z^2 - 6z + 9} \right) = 2\left( \sqrt{x^2 - 26x + 169 + y^2 + 6y + 9 + z^2 - 26z + 169} \right)\]
\[\text{ Squaring both sides }, \]
\[ \Rightarrow 9\left( x^2 + y^2 + z^2 + 4x - 4y - 6z + 17 \right) = 4\left( x^2 + y^2 + z^2 - 26x + 6y - 26z + 347 \right)\]
\[ \Rightarrow 9 x^2 + 9 y^2 + 9 z^2 + 36x - 36y - 54z + 153 = 4 x^2 + 4 y^2 + 4 z^2 - 104x + 24y - 104z + 1388\]
\[ \Rightarrow 5 x^2 + 5 y^2 + 5 z^2 + 140x - 60y + 50z - 1235 = 0\]
\[ \Rightarrow 5\left( x^2 + y^2 + z^2 \right) + 140x - 60y + 50z - 1235 = 0\]
\[ \therefore 5\left( x^2 + y^2 + z^2 \right) + 140x - 60y + 50z - 1235 = 0 \text{ is the locus of the point P } .\]
APPEARS IN
संबंधित प्रश्न
Coordinate planes divide the space into ______ octants.
If the origin is the centroid of the triangle PQR with vertices P (2a, 2, 6), Q (–4, 3b, –10) and R (8, 14, 2c), then find the values of a, b and c.
Name the octants in which the following points lie: (5, 2, 3)
Name the octants in which the following points lie:
(–5, 4, 3)
Name the octants in which the following points lie:
(4, –3, 5)
Name the octants in which the following points lie:
(–5, –3, –2)
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 points on z-axis which are at a distance \[\sqrt{21}\]from the point (1, 2, 3).
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.
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.
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).
Are the points A(3, 6, 9), B(10, 20, 30) and C(25, –41, 5), the vertices of a right-angled triangle?
Verify the following:
(0, 7, 10), (–1, 6, 6) and (–4, 9, –6) are vertices of a right-angled triangle.
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.
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
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
XOZ-plane divides the join of (2, 3, 1) and (6, 7, 1) in the ratio
The coordinates of the foot of the perpendicular drawn from the point P(3, 4, 5) on the yz- plane are
The coordinates of the foot of the perpendicular from a point P(6,7, 8) on x - axis are
The perpendicular distance of the point P (6, 7, 8) from xy - plane is
The length of the perpendicular drawn from the point P (3, 4, 5) on y-axis is
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.
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 angle between the lines whose direction cosines are given by the equations l + m + n = 0, l2 + m2 – n2 = 0
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.
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
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 plane 2x – 3y + 6z – 11 = 0 makes an angle sin–1(α) with x-axis. The value of α is equal to ______.
The line `vecr = 2hati - 3hatj - hatk + lambda(hati - hatj + 2hatk)` lies in the plane `vecr.(3hati + hatj - hatk) + 2` = 0.
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.
