Advertisements
Advertisements
प्रश्न
If the points A(−2, 1), B(a, b) and C(4, −1) ae collinear and a − b = 1, find the values of aand b.
Advertisements
उत्तर
The given points A(−2, 1), B(a, b) and C(4, −1) are collinear.
\[\therefore \text{ ar } \left( ∆ ABC \right) = 0\]
\[ \Rightarrow \frac{1}{2}\left| x_1 \left( y_2 - y_3 \right) + x_2 \left( y_3 - y_1 \right) + x_3 \left( y_1 - y_2 \right) \right| = 0\]
\[ \Rightarrow x_1 \left( y_2 - y_3 \right) + x_2 \left( y_3 - y_1 \right) + x_3 \left( y_1 - y_2 \right) = 0\]
\[\Rightarrow - 2\left[ b - \left( - 1 \right) \right] + a\left( - 1 - 1 \right) + 4\left( 1 - b \right) = 0\]
\[ \Rightarrow - 2b - 2 - 2a + 4 - 4b = 0\]
\[ \Rightarrow - 2a - 6b = - 2\]
\[ \Rightarrow a + 3b = 1 . . . . . \left( 1 \right)\]
Also, it is given that
a − b = 1 .....(2)
Solving (1) and (2), we get
\[b + 1 + 3b = 1\]
\[ \Rightarrow 4b = 0\]
\[ \Rightarrow b = 0\]
Putting b = 0 in (1), we get
\[a + 3 \times 0 = 1\]
\[ \Rightarrow a = 1\]
Hence, the respective values of a and b are 1 and 0.
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
If two opposite vertices of a square are (5, 4) and (1, −6), find the coordinates of its remaining two vertices.
Prove that the points (0, 0), (5, 5) and (-5, 5) are the vertices of a right isosceles triangle.
Find the coordinates of the point which divides the line segment joining (−1,3) and (4, −7) internally in the ratio 3 : 4
Find the points of trisection of the line segment joining the points:
5, −6 and (−7, 5),
If three consecutive vertices of a parallelogram are (1, -2), (3, 6) and (5, 10), find its fourth vertex.
If the point P (2,2) is equidistant from the points A ( -2,K ) and B( -2K , -3) , find k. Also, find the length of AP.
Show that the following points are the vertices of a rectangle
A (0,-4), B(6,2), C(3,5) and D(-3,-1)
The base QR of a n equilateral triangle PQR lies on x-axis. The coordinates of the point Q are (-4, 0) and origin is the midpoint of the base. Find the coordinates of the points P and R.
Prove that the diagonals of a rectangle ABCD with vertices A(2,-1), B(5,-1) C(5,6) and D(2,6) are equal and bisect each other
The abscissa of any point on y-axis is
The distance of the point P (4, 3) from the origin is
Write the distance between the points A (10 cos θ, 0) and B (0, 10 sin θ).
If the distance between the points (3, 0) and (0, y) is 5 units and y is positive. then what is the value of y?
If x is a positive integer such that the distance between points P (x, 2) and Q (3, −6) is 10 units, then x =
If points (a, 0), (0, b) and (1, 1) are collinear, then \[\frac{1}{a} + \frac{1}{b} =\]
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
If the centroid of the triangle formed by (7, x) (y, −6) and (9, 10) is at (6, 3), then (x, y) =
If the points P (x, y) is equidistant from A (5, 1) and B (−1, 5), then
Assertion (A): The ratio in which the line segment joining (2, -3) and (5, 6) internally divided by x-axis is 1:2.
Reason (R): as formula for the internal division is `((mx_2 + nx_1)/(m + n) , (my_2 + ny_1)/(m + n))`
