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Let the coordinate axes be inclined at an angle θ. Then the distance between two points P(x_{1}, y_{1}) and Q(x_{2}, y_{2}) is given by
Note that if the axes are inclined at 90° , then the distance is given by
The point equidistant from (0, 0), (32,10) and (42,0) is
Proof: LetP (h,k) be the required point which is equidistant from 0(0,0). A (32,10) and B(42,0).
Then
PO=PA and PO= PB
Now
Important Rules
R  1: If the coordinates of three vertices of a triangle are given then it represents a Right angled Triangle if the product of the slopes of two adjacent sides is1.
OR
The square of the largest side equals the sum of the squares of the other two sides.
R  2: If the coordinates of four vertices of a quadrilateral are known, then it represents a
(i) Rectangle : If
(ii) Square : if
(iii) Parallelogram : if any one of the following conditions is true
(iv) Rhombus : if
Proof: Let A(0, 1), B(2, 3), C(6, 7) and D(8, 3) be the given point. Then using the distance formula, we get
If (x,y) is a point equidiantant from (3,4) and (1,2), then
Proof:It is given that the point P(x, y) is equidistant from the points A(3,4)and B(1,2).
i.e PAPB
The coordinates of point P(x, y) dividing the line segment joining two points A(x_{1} , y_{1}) and B(x_{2}, y_{2}) joining the ratio m: n internally arc given by
The coordinates of a point P(x, y) which divides the line segment joining A(x_{1}, y_{1}) and B(x_{2}, y_{2}) externally in the ratio m: n, are given by
Comments about Centroid of a Triangle.
The centroid of the triangle, whose vertices are (3, 5), (7, 4), (10,2), is given by
The coordinates of the centroid will be
The point dividing the join of (2, 1) and (3, 5) externally in the ratio 2 : 3 is given by
The area of a triangle with vertices as (x_{1}, y_{1}), (x_{2}, y_{2}) and (x_{3}, y_{3}) is given
The Three coordinate planes divide the whole space into how many octants?
The three coordinate planes divide the whole space into Eight Parts, called octants. These eight octants and signs of the coordinates of a point P in each one of them are shown below :
Remark : Three fixed lines X'OX Y' O Y and Z' O Z are called x, y and zaxis respectively and their common point of intersection ‘O' is called the origin or pole of coordinates.
The perpendicular distance of any point P(x,y, z) from the zaxis is given by
We have
1. Distance between two points (x_{1}, y_{1}, z_{1}) and
2. Distance of a point (x_{1}, y_{1}, z_{1}) from the origin
3. Distance of a point (x_{1}, y_{1}, z_{1}) from xaxis
4. Distance of a point (x_{1}, y_{1}, z_{1}) from yaxis
5. Distance of a point (x_{1}, y_{1}, z_{1}) from zaxis
Note: We have
Remark: In particular, the equations of YZ, ZX and XY planes are x = 0, y = 0 and z = 0 respectively.
What is the locus of a point for which x = a and y= b?
x = a is a plane   to YZ plane
y = b is a plane   to ZX plane
Therefore x = a and y = b is a line parallel to zaxis
Note : Remember that two planes intersect in a line
What are the length of the edges of the rectangular parallolopiped formed by planes drawn through the point (a. b, c) and (p, q, r) parallel to the coordinate planes?
Lengths of the edges will be
p  a, q  b and r  c
Which of the following figures is formed by the points whose coordinates are (0,7,10), (1,6,6), (4,9,6)?
The distance between two points in a three dimensional can be calculated as :
d = [(x_{2}x_{1})^{2} + (y_{2}y_{1})^{2} + (z_{2}z_{1})^{2}]^{½}
The coordinates are P(0,7,10), Q(1,6,6), R(4,9,6)
PQ = [(0+1)^{2 }+ (76)^{2} + (106)^2]^{½}
PQ = [1+1+16]^{½}
PQ = [18]^{½}
QR = [(14)^{2} + (69)^{2 }+ (66)^2]^{½}
QR = [9+9+0]
QR = [18]^{½}
PR = [(0+4)^{2} + (79)^{2} + (106)^2]^{½}
PR = [16 + 4 + 16]^{½}
PR = 6
As we can see that (PR)^{2} = (QR)^{2} + (PQ)^{2}
I.e. isoceles and right triangle
The point (4,3,6) lies in which of the eight octants? (x',y',z', are negative version of x,y,z)
What is radius of a sphere passing through the four (3,2,2), (1,1,3), (0,5,6), (2,1,2) and having centre (1,3,4)?
Centre o f the sphere = ( 1 , 3 , 4 )
Sphere passes through (3. 2, 2)
∴ Radius
Remark: Sphere passes through four points.
Note: Distance of any one of the ratio m : n arc given as follows
The YZ plane divides the line joining the points (2, 4. 5), (3, 5, 4) in the ratio
The coordinates (x, y, z) of a point P which divides the line joining A(x_{1}, y_{1} z_{1} ) and B(x_{2}, y_{2}, z_{2}) in the ratio m : n are given as follows:
1. P divides AB internally in the ratio m : n
Then
2. P divides AB externally in the ratio m : n
Now yz plane is given by x = 0. It means that suppose the point whose xcoordinate is zero divides the line joining (2, 4, 5) and (3. 5, 6) in the ratio m : n.
Then
What are the coordinates of the centroid of the triangle whose vertices are (3, 2, 0), (5, 3, 2} and (9, 6, 3)?
Centroid of the triangle whose vertices are (x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}), (x_{3}, y_{3}, z_{3}) are given by
In this question, the vertices of the triangle are (3, 2, 0), (5, 3. 2) and (9, 6, 3).
∴The coordinates of the centroid are
About Tetrahedron
1. Tetrahedron is a pyramid with a triangular base.
2. It has four vertices.
3. It has four faces. If BCD is the triangular base and A the top ertex, then the four faces are BCD, ABC, ACDand ABD
4. It has six edges, AB, AC, AD, BC, CD and DB are its six edges.
5. If (x_{1},y_{1},z_{1}) , (x_{2},y_{2},z_{2}), (x_{3},y_{3},z_{3}) and (x_{4},y_{4},z_{4}) are the coordinates of the vertices, then the Coordinates of its centroid (centre of gravity) are
above figure shows a tetrahedron. Rase triangle is BCD. The centre of gravity G of the tetrahedron divides the line AG. in the ratio 3:1, where G, is the centre of gravity of the base triangle BC (E is the middle point of DC and Gt divides BE in :he ratio 2 : 1).
6. The volume V of the tetrahedron is given by
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