Advertisements
Advertisements
Question
If three points (x1, y1) (x2, y2), (x3, y3) lie on the same line, prove that \[\frac{y_2 - y_3}{x_2 x_3} + \frac{y_3 - y_1}{x_3 x_1} + \frac{y_1 - y_2}{x_1 x_2} = 0\]
Advertisements
Solution
GIVEN: If three points (x1, y1) (x2, y2) and (x3, y3) lie on the same line
TO PROVE: \[\frac{y_2 - y_3}{x_2 x_3} + \frac{y_3 - y_1}{x_3 x_1} + \frac{y_1 - y_2}{x_1 x_2} = 0\]
PROOF:
We know that three points (x1, y1) (x2, y2) and (x3, y3) are collinear if
`x_1 (y_2 - y_3) + x_2 (y_3 - y_1) + x_3 (y_1 - y_2 ) = 0`
⇒ `x_1 (y_2 - y_3) + x_2 (y_3 - y_1) + x_3 (y_1 - y_2 ) = 0`
Dividing by `x_1 x_2 x_3`
⇒ \[\frac{x_1 (y_2 - y_3 ) }{x_1 x_2 x_3} + \frac{x_2 (y_3 - y_1 ) }{x_1x_2 x_3} + \frac{x_3 ( y_1 - y_2 ) }{x_1 x_2 x_3} = 0\]
⇒ \[\frac{(y_2 - y_3)}{x_2 x_3} + \frac{(y_3 - y_1)}{x_3 x_1} + \frac{(y_1 - y_2)}{x_1 x_2} = 0\]
Hence proved.
APPEARS IN
RELATED QUESTIONS
Let ABCD be a square of side 2a. Find the coordinates of the vertices of this square when The centre of the square is at the origin and coordinate axes are parallel to the sides AB and AD respectively.
If G be the centroid of a triangle ABC, prove that:
AB2 + BC2 + CA2 = 3 (GA2 + GB2 + GC2)
Prove that the points (3, 0), (4, 5), (-1, 4) and (-2, -1), taken in order, form a rhombus.
Also, find its area.
Find the points of trisection of the line segment joining the points:
(3, -2) and (-3, -4)
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
Prove that (4, 3), (6, 4) (5, 6) and (3, 5) are the angular points of a square.
In what ratio does the point (−4, 6) divide the line segment joining the points A(−6, 10) and B(3,−8)?
Show that the following points are the vertices of a square:
A (6,2), B(2,1), C(1,5) and D(5,6)
Find the value of a, so that the point ( 3,a ) lies on the line represented by 2x - 3y =5 .
A point whose abscissa is −3 and ordinate 2 lies in
Two points having same abscissae but different ordinate lie on
Show that the points (−4, −1), (−2, −4) (4, 0) and (2, 3) are the vertices points of a rectangle.
If the distance between points (x, 0) and (0, 3) is 5, what are the values of x?
Write the ratio in which the line segment doining the points A (3, −6), and B (5, 3) is divided by X-axis.
Find the distance between the points \[\left( - \frac{8}{5}, 2 \right)\] and \[\left( \frac{2}{5}, 2 \right)\] .
What is the nature of the line which includes the points (-5, 5), (6, 5), (-3, 5), (0, 5)?
The point R divides the line segment AB, where A(−4, 0) and B(0, 6) such that AR=34AB.">AR = `3/4`AB. Find the coordinates of R.
Abscissa of all the points on the x-axis is ______.
Ryan, from a very young age, was fascinated by the twinkling of stars and the vastness of space. He always dreamt of becoming an astronaut one day. So, he started to sketch his own rocket designs on the graph sheet. One such design is given below :
Based on the above, answer the following questions:
i. Find the mid-point of the segment joining F and G. (1)
ii. a. What is the distance between the points A and C? (2)
OR
b. Find the coordinates of the points which divides the line segment joining the points A and B in the ratio 1 : 3 internally. (2)
iii. What are the coordinates of the point D? (1)
Assertion (A): Mid-point of a line segment divides the line segment in the ratio 1 : 1
Reason (R): The ratio in which the point (−3, k) divides the line segment joining the points (− 5, 4) and (− 2, 3) is 1 : 2.
