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If the points A(6, 1), B(8, 2) C(9, 4) and D(P, 3) are the vertices of a parallelogram taken in order find the value of P.
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Mid point of
Mid point of
But
Find the co–ordinates of the points of trisection of the line segment joining the points A(2,–2) and B(–7, 4).
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Ratio
Check whether the points (2, 2), (4, 0) and (–6, 10) are collinear.
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So the points are collinear
Find the ratio in which the yaxis divides the join of (5, – 6) and (–1, – 4).
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Let the ratio be
Point
Ratio
Find the coordinates of a point A, where AB is the diameter of a circle whose centre is O(2, –3) and B is (1, 4).
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Let be
Using section formula, show that the points A(–3, 1), B (1, 3) and C(–1, 1) are collinear.
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Let the ratio be
using section formula
…(1)
and ….(2)
From (1)
divides in the ratio that is is the mid point of
So are collinear
Find the value of p, for which the points (1, 3), (3, p) and (5, –1) are collinear.
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Find points on the xaxis, which are at a distance of 5 units from the point A (5, –3).
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xaxis
Points on xaxis and
Show that the points (a, b + c), (b, c + a) and (c, a + b) are collinear.
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Area of
Taking common from ,
Hence points are collinear
Prove that the points (0, 0), (5, 5) and (–5, 5) are the vertices of a right angled isosceles triangle.
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So it is right angled isosceles triangle.
Find the value of x, if the distance between the points (x, –1) and (3, –2) is x + 5.
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If the point C (–1, 2) divides internally the line segment joining A (2, 5) and B(x, y) in the ratio 3 : 4, then find the coordinates of B.
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Show that the point P (–4, 2) lies on the line segment joining the points A (– 4, 6) and B (– 4, –6 ) .
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since
Hence lies on the line segment
Show that the point (1, –1) is the centre of the circle circumscribing the triangle whose vertices are (4, 3) and (–2, 3) and (6, –1).
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Let be the centre of the circle.
is the centre of the circle
If A(1, 2), B (4, y), C(x, 6) and D(3, 5) are the vertices of a parallelogram ABCD taken in order, find the values of x and y.
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Mid point of mid point of
and
If the point P(x, y) is equidistant from the points A(5, 1) and B (–1, 5) then prove that 3x = 2y.
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(Proved)
Determine the ratio in which the point P(x, –2) divides the join of A(–4, 3) and B(2, –4). Also find the value of x.
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Let the ratio be
But
If points A(–2, –1), B (a, 0), C (4, b) and D(1, 2) are the vertices of a parallelogram ABCD, find the values of a and b.
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Mid point of Mid point of
Find the value of x such that PQ = QR, where the coordinates of P, Q and R are (6, –1), (1, 3) and (x, 8) respectively.
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Find the value of k for which the point (8, 1),(k, –4) and (2, –5) are collinear.
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Show that P(1, –1) is the centre of the circle circumscribing the triangle whose angular points are A (4, 3), B(–2, 3) and C (6, –1).
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Since units, is the centre of the circle.
One end of a diameter of a circle is at (2, 3) and the centre is (–2, 5). What are the cooridnates of the other end of the diameter?
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and
and
and
The other end is
A point P is at a distance of \(\sqrt{10} \) from the point (2, 3). Find the coordinates of the point P if its y coordinate is twice its x coordinate.
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If then and if then
Find the coordinates of the point B, if the point P(– 4, 1) divides the line segment joining the points A(2, –2) and B in the ratio 3 : 5.
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Find the third vertex of the triangle ABC if two of its vertices are at A(–3, 1) and B (0, 2) and the midpoint of BC is at \(D\ \left(\dfrac{3}{2},\dfrac{1}{2}\right) \)
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And
Find the value of s if the point P(0, 2) is equidistant from Q (3, s) and R(s, 5).
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Find the perimeter of the triangle formed by the points (0, 0), (1, 0), (0, 1).
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Find the points on the xaxis which are at a distnace of \(2\sqrt5 \) from the point \((7,4) \). How many such points are there?
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Point on xaxis be
The points are
If \(A \) and \(B \) are the points (–2, –2) and (2, –4) respectively, find the coordinates of P on the line segment AB such that \(AP=\dfrac{3}{7}\ AB \).
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Coordinates of
Find a point on xaxis which is equidistant from A(–3, 4) and B(1, –4).
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As
Find the value of a so that the point (3, a) lies on the line represented by 2x – 3y = 5.
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The points \(A(x_1,y_1),\ B(x_2,y_2) \) and \(C(x_3,y_3) \) are the vertices of \(\triangle ABC \)
(i) The median from A meets BC at D. What are the coordinates of the point D?
(ii) Find the coordinates of the point P on AD such that AP : PD = 2 : 1.
(iii) Find the coordinates of points Q and R on medians BE and CF respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1.
(iv) What are the coordinates of the centroid of the triangle ABC?
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(i) Coordinates of BC mid point
(ii) Let the coordinates of a point be and ratio is
Coordinate of
(iii) Let the coordinates of be and ratio is
Coordinates of
Mid point of AC = coordinate of E
So the required coordinate of Q
Coordinate of
Mid point of AB = Coordinate of F
(iv) Coordinate of the centroid
If P and Q are two points whose coordinates are \((at^2,2at) \) and \(\left(\dfrac{a}{t^2},\dfrac{2a}{t}\right) \) respectively and S is the point (a, 0), show that \(\dfrac{1}{SP}+\dfrac{1}{SQ} \) is independent of t.
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…(1)
If (– 4, 3) and (4, 3) are two vertices of an equilateral triangle, find the coordinates of the third vertex given that the origin lies in the interior of the triangle.
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Distance between and …(1)
Distance between and
…(2)
Distance between (4, 3) and (4,3)
Eq (1) = Eq (2)
Eq (1)
…(3)
Substituting the value of x in 3
Neglect as if then origin cannot interior of triangle
Therefore the third vetrex
If the points \((x,y),\ (x_1,y_1) \) and \((xx_1,yy_1) \) are collinear, show that \(xy_1=x_1y \). Also, show that the line joining the given points passes through the origin.
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(Proved)
Mr. Aggarwal starts walking from his home to office. Instead of going to the office directly, he goes to a bank first, from there to his daughter’s school and then reaches the office. What is the extra distance travelled by Mr. Aggarwal in reaching his office ? Assume that all distances covered are in straight lines. If the house is situated at (2, 4), bank at (5, 8), school at (13, 14) and office at (13, 26) and coordinates are in km.
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The position of agarwal’s house is and position of bank is
Distance between the house and the bank
units
The position of bank is and position of school
Distance between the bank and the school
The distance between the bank and the school
Let the total distance
Let be the shortest distance
Extra distance
Find the centre of a circle passing through the points (6, –6), (3, –7) and (3, 3).
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…(1)
Also
…(2)
From (1) and (2) we get
centre