Advertisements
Advertisements
Question
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.
Solution
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
RELATED QUESTIONS
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:
(7, 4, –3)
Find the point on y-axis which is equidistant from the points (3, 1, 2) and (5, 5, 2).
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.
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).
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:
(0, 7, –10), (1, 6, –6) and (4, 9, –6) are vertices of an isosceles triangle.
Verify the following:
(0, 7, 10), (–1, 6, 6) and (–4, 9, –6) are vertices of a right-angled triangle.
Verify the following:
(–1, 2, 1), (1, –2, 5), (4, –7, 8) and (2, –3, 4) are vertices of a parallelogram.
Write the distance of the point P(3, 4, 5) from z-axis.
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 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
What is the locus of a point for which y = 0, z = 0?
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(3, 3,4) from the x-axis is
Find the image of the point having position vector `hati + 3hatj + 4hatk` in the plane `hatr * (2hati - hatj + hatk)` + 3 = 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 an angle of `pi/4` with each of y and z axis, then the angle which it makes with x-axis is ______.
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
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 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.
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 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 intercepts made by the plane 2x – 3y + 5z +4 = 0 on the co-ordinate axis are `-2, 4/3, - 4/5`.
