Advertisements
Advertisements
Question
If the points (k, 2k), (3k, 3k) and (3, 1) are collinear, then k
Options
- \[\frac{1}{3}\]
- \[- \frac{1}{3}\]
- \[\frac{2}{3}\]
- \[- \frac{2}{3}\]
Solution
We have three collinear points A(k, 2k) : B(3k, 3k) : C (3, 1) .
In general if `A(x_1 , y_1) ; B (x_2 ,y_2) ; C(x_3 ,y_3) `are collinear then, area of the triangle is 0.
`x_1 (y_2 - y_3 ) + x_2 (y_3 - y_1) + x_3 (y_1- y_2 ) = 0`
So,
`k (3k - 1 ) + 3k(1 - 2k) + 3 (2k - 3k) = 0`
So,
-3k2 - k = 0
Take out the common terms,
-k (3k + 1) = 0
Therefore,
`k = -1/3`
APPEARS IN
RELATED QUESTIONS
On which axis do the following points lie?
R(−4,0)
If two opposite vertices of a square are (5, 4) and (1, −6), find the coordinates of its remaining two vertices.
Name the quadrilateral formed, if any, by the following points, and given reasons for your answers:
A(4, 5) B(7, 6), C (4, 3), D(1, 2)
Find the coordinates of the point where the diagonals of the parallelogram formed by joining the points (-2, -1), (1, 0), (4, 3) and(1, 2) meet
The line segment joining the points P(3, 3) and Q(6, -6) is trisected at the points A and B such that Ais nearer to P. If A also lies on the line given by 2x + y + k = 0, find the value of k.
If the point C ( - 2,3) is equidistant form the points A (3, -1) and Bx (x ,8) , find the value of x. Also, find the distance between BC
Show that the points A(2,1), B(5,2), C(6,4) and D(3,3) are the angular points of a parallelogram. Is this figure a rectangle?
Show hat A(1,2), B(4,3),C(6,6) and D(3,5) are the vertices of a parallelogram. Show that ABCD is not rectangle.
Show that the following points are the vertices of a rectangle
A (0,-4), B(6,2), C(3,5) and D(-3,-1)
ΔXYZ ∼ ΔPYR; In ΔXYZ, ∠Y = 60o, XY = 4.5 cm, YZ = 5.1 cm and XYPY =` 4/7` Construct ΔXYZ and ΔPYR.
The measure of the angle between the coordinate axes is
Points P, Q, R and S divides the line segment joining A(1, 2) and B(6, 7) in 5 equal parts. Find the coordinates of the points P, Q and R.
Find the value of k if points A(k, 3), B(6, −2) and C(−3, 4) are collinear.
If P (2, 6) is the mid-point of the line segment joining A(6, 5) and B(4, y), find y.
If the centroid of a triangle is (1, 4) and two of its vertices are (4, −3) and (−9, 7), then the area of the triangle is
The coordinates of the circumcentre of the triangle formed by the points O (0, 0), A (a, 0 and B (0, b) are
Which of the points P(-1, 1), Q(3, - 4), R(1, -1), S (-2, -3), T(-4, 4) lie in the fourth quadrant?
Which of the points P(0, 3), Q(1, 0), R(0, –1), S(–5, 0), T(1, 2) do not lie on the x-axis?
Find the coordinates of the point whose abscissa is 5 and which lies on x-axis.
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?
