Advertisements
Advertisements
प्रश्न
Find the value of k, if the points A (8, 1) B(3, −4) and C(2, k) are collinear.
Advertisements
उत्तर
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
संबंधित प्रश्न
If the points A(k + 1, 2k), B(3k, 2k + 3) and C(5k − 1, 5k) are collinear, then find the value of k
Find the ratio in which the point (2, y) divides the line segment joining the points A (-2,2) and B (3, 7). Also, find the value of y.
If the coordinates of the mid-points of the sides of a triangle be (3, -2), (-3, 1) and (4, -3), then find the coordinates of its vertices.
If A and B are (1, 4) and (5, 2) respectively, find the coordinates of P when AP/BP = 3/4.
If the poin A(0,2) is equidistant form the points B (3, p) and C (p ,5) find the value of p. Also, find the length of AB.
If (2, p) is the midpoint of the line segment joining the points A(6, -5) and B(-2,11) find the value of p.
ABCD is rectangle formed by the points A(-1, -1), B(-1, 4), C(5, 4) and D(5, -1). If P,Q,R and S be the midpoints of AB, BC, CD and DA respectively, Show that PQRS is a rhombus.
Find the area of quadrilateral PQRS whose vertices are P(-5, -3), Q(-4,-6),R(2, -3) and S(1,2).
If the point A(0,2) is equidistant from the points B(3,p) and C(p, 5), find p.
If the point C(k,4) divides the join of A(2,6) and B(5,1) in the ratio 2:3 then find the value of k.
Find the coordinates of the centre of the circle passing through the points P(6, –6), Q(3, –7) and R (3, 3).
Two points having same abscissae but different ordinate lie on
Show that ΔABC, where A(–2, 0), B(2, 0), C(0, 2) and ΔPQR where P(–4, 0), Q(4, 0), R(0, 2) are similar triangles.
Find the value(s) of k for which the points (3k − 1, k − 2), (k, k − 7) and (k − 1, −k − 2) 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 points A (1,2) , O (0,0) and C (a,b) are collinear , then find a : b.
What is the nature of the line which includes the points (-5, 5), (6, 5), (-3, 5), (0, 5)?
The line segment joining the points A(2, 1) and B (5, - 8) is trisected at the points P and Q such that P is nearer to A. If P also lies on the line given by 2x - y + k= 0 find the value of k.
Point P(– 4, 2) lies on the line segment joining the points A(– 4, 6) and B(– 4, – 6).
Find the coordinates of the point which lies on x and y axes both.
