Advertisements
Advertisements
Question
In what ratio does the point (−4, 6) divide the line segment joining the points A(−6, 10) and B(3,−8)?
Advertisements
Solution
The co-ordinates of a point which divided two points `(x_1,y_1)` and `(x_2, x_2)` internally in the ratio m:n is given by the formula,
`(x,y) = (((mx_2 + nx_1)/(m + n))","((my_2 + ny_1)/(m + n)))`
Here it is said that the point (−4,6) divides the points A(−6,10) and B(3,−8). Substituting these values in the above formula we have,
`(-4, 6) = (((m(3) + n(-6))/(m + n))"," ((m(-8) + n(10))/
(m + n)))`
Equating the individual components we have,
`-4 = (m(3) + n(-6))/(m + n)`
-4m - 4n = 3m - 6n
7m = 2n
`m/n = 2/7`
Therefore the ratio in which the line is divided is 2 : 7
RELATED QUESTIONS
A line intersects the y-axis and x-axis at the points P and Q respectively. If (2, –5) is the mid-point of PQ, then find the coordinates of P and Q.
Prove that the points (−2, 5), (0, 1) and (2, −3) are collinear.
Find the equation of the perpendicular bisector of the line segment joining points (7, 1) and (3,5).
Prove that (4, 3), (6, 4) (5, 6) and (3, 5) are the angular points of a square.
Find the ratio which the line segment joining the pints A(3, -3) and B(-2,7) is divided by x -axis Also, find the point of division.
Find the coordinates of the centre of the circle passing through the points P(6, –6), Q(3, –7) and R (3, 3).
Find the possible pairs of coordinates of the fourth vertex D of the parallelogram, if three of its vertices are A(5, 6), B(1, –2) and C(3, –2).
Find the area of the quadrilateral ABCD, whose vertices are A(−3, −1), B (−2, −4), C(4, − 1) and D (3, 4).
The abscissa of any point on y-axis is
If A (1, 2) B (4, 3) and C (6, 6) are the three vertices of a parallelogram ABCD, find the coordinates of fourth vertex D.
The coordinates of the point on X-axis which are equidistant from the points (−3, 4) and (2, 5) are
The ratio in which the x-axis divides the segment joining (3, 6) and (12, −3) is
The distance of the point (4, 7) from the x-axis is
If P is a point on x-axis such that its distance from the origin is 3 units, then the coordinates of a point Q on OY such that OP = OQ, are
If the centroid of the triangle formed by the points (3, −5), (−7, 4), (10, −k) is at the point (k −1), then k =
The coordinates of the fourth vertex of the rectangle formed by the points (0, 0), (2, 0), (0, 3) are
Find the point on the y-axis which is equidistant from the points (S, - 2) and (- 3, 2).
The line 3x + y – 9 = 0 divides the line joining the points (1, 3) and (2, 7) internally in the ratio ______.
(–1, 7) is a point in the II quadrant.
A tiling or tessellation of a flat surface is the covering of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. Historically, tessellations were used in ancient Rome and in Islamic art. You may find tessellation patterns on floors, walls, paintings etc. Shown below is a tiled floor in the archaeological Museum of Seville, made using squares, triangles and hexagons.
A craftsman thought of making a floor pattern after being inspired by the above design. To ensure accuracy in his work, he made the pattern on the Cartesian plane. He used regular octagons, squares and triangles for his floor tessellation pattern
Use the above figure to answer the questions that follow:
- What is the length of the line segment joining points B and F?
- The centre ‘Z’ of the figure will be the point of intersection of the diagonals of quadrilateral WXOP. Then what are the coordinates of Z?
- What are the coordinates of the point on y-axis equidistant from A and G?
OR
What is the area of Trapezium AFGH?
