Advertisements
Advertisements
Question
Find the value of k, if the points A (8, 1) B(3, −4) and C(2, k) are collinear.
Solution
The formula for the area ‘A’ encompassed by three points( x1 , y1) , (x2 , y2) and (x3 , y3) is given by the formula,
\[∆ = \frac{1}{2}\left| \left( x_1 y_2 + x_2 y_3 + x_3 y_1 \right) - \left( x_2 y_1 + x_3 y_2 + x_1 y_3 \right) \right|\]
If three points are collinear the area encompassed by them is equal to 0.
The three given points are A(8,1), B(3,−4) and C(2,k). It is also said that they are collinear and hence the area enclosed by them should be 0.
\[∆ = \frac{1}{2}\left| \left( 8 \times - 4 + 3 \times k + 2 \times 1 \right) - \left( 3 \times 1 + 2 \times - 4 + 8 \times k \right) \right|\]
\[ 0 = \frac{1}{2}\left| \left( - 32 + 3k + 2 \right) - \left( 3 - 8 + 8k \right) \right|\]
\[ 0 = \frac{1}{2}\left| - 25 - 5k \right|\]
\[k = - 5\]
Hence the value of ‘k’ for which the given points are collinear is k = - 5.
APPEARS IN
RELATED QUESTIONS
Prove that the points (−2, 5), (0, 1) and (2, −3) are collinear.
If two opposite vertices of a square are (5, 4) and (1, −6), find the coordinates of its remaining two vertices.
Find the coordinates of the circumcentre of the triangle whose vertices are (3, 0), (-1, -6) and (4, -1). Also, find its circumradius.
If the vertices of ΔABC be A(1, -3) B(4, p) and C(-9, 7) and its area is 15 square units, find the values of p
ABCD is a rectangle whose three vertices are A(4,0), C(4,3) and D(0,3). Find the length of one its diagonal.
Find the point on x-axis which is equidistant from points A(-1,0) and B(5,0)
Show that the points (−2, 3), (8, 3) and (6, 7) are the vertices of a right triangle ?
If (a,b) is the mid-point of the line segment joining the points A (10, - 6) , B (k,4) and a - 2b = 18 , find the value of k and the distance AB.
In \[∆\] ABC , the coordinates of vertex A are (0, - 1) and D (1,0) and E(0,10) respectively the mid-points of the sides AB and AC . If F is the mid-points of the side BC , find the area of \[∆\] DEF.
If the vertices of a triangle are (1, −3), (4, p) and (−9, 7) and its area is 15 sq. units, find the value(s) of p.
If three points (x1, y1) (x2, y2), (x3, y3) lie on the same line, prove that \[\frac{y_2 - y_3}{x_2 x_3} + \frac{y_3 - y_1}{x_3 x_1} + \frac{y_1 - y_2}{x_1 x_2} = 0\]
If the points A(−1, −4), B(b, c) and C(5, −1) are collinear and 2b + c = 4, find the values of b and c.
Write the ratio in which the line segment joining points (2, 3) and (3, −2) is divided by X axis.
Write the condition of collinearity of points (x1, y1), (x2, y2) and (x3, y3).
What is the distance between the points A (c, 0) and B (0, −c)?
The coordinates of the fourth vertex of the rectangle formed by the points (0, 0), (2, 0), (0, 3) are
If A(4, 9), B(2, 3) and C(6, 5) are the vertices of ∆ABC, then the length of median through C is
The distance of the point P(2, 3) from the x-axis is ______.
Statement A (Assertion): If the coordinates of the mid-points of the sides AB and AC of ∆ABC are D(3, 5) and E(–3, –3) respectively, then BC = 20 units.
Statement R (Reason): The line joining the mid-points of two sides of a triangle is parallel to the third side and equal to half of it.
The distance of the point (–6, 8) from x-axis is ______.
