Advertisements
Advertisements
Question
If R (x, y) is a point on the line segment joining the points P (a, b) and Q (b, a), then prove that x + y = a + b.
Advertisements
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.
It is said that the point R(x, y) lies on the line segment joining the points P(a, b) and Q(b, a). Hence we understand that these three points are collinear. So the area enclosed by them should be 0.
\[∆ = \frac{1}{2}\left| \left( ay + xa + b^2 \right) - \left( xb + by + a^2 \right) \right|\]
\[ 0 = \frac{1}{2}\left| ay + xa + b^2 - xb - by - a^2 \right|\]
\[ 0 = ay + xa + b^2 - xb - by - a^2 \]
\[ a^2 - b^2 = ax + ay - bx - by\]
\[ \left( a + b \right)\left( a - b \right) = \left( a - b \right)\left( x + y \right)\]
\[ \left( a + b \right) = \left( x + y \right)\]
Hence under the given conditions we have proved that x + y = a + b .
APPEARS IN
RELATED QUESTIONS
Find the distance between the following pair of points:
(a, 0) and (0, b)
Find the coordinates of the circumcentre of the triangle whose vertices are (3, 0), (-1, -6) and (4, -1). Also, find its circumradius.
Find a point on the x-axis which is equidistant from the points (7, 6) and (−3, 4).
Show that the points A(5, 6), B(1, 5), C(2, 1) and D(6,2) are the vertices of a square.
Find the points of trisection of the line segment joining the points:
(2, -2) and (-7, 4).
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.
Point P(x, 4) lies on the line segment joining the points A(−5, 8) and B(4, −10). Find the ratio in which point P divides the line segment AB. Also find the value of x.
Mark the correct alternative in each of the following:
The point of intersect of the coordinate axes is
The measure of the angle between the coordinate axes is
Show that the points (−4, −1), (−2, −4) (4, 0) and (2, 3) are the vertices points of a rectangle.
Prove hat the points A (2, 3) B(−2,2) C(−1,−2), and D(3, −1) are the vertices of a square ABCD.
If A(−3, 5), B(−2, −7), C(1, −8) and D(6, 3) are the vertices of a quadrilateral ABCD, find its area.
Write the formula for the area of the triangle having its vertices at (x1, y1), (x2, y2) and (x3, y3).
If points (t, 2t), (−2, 6) and (3, 1) are collinear, then t =
The coordinates of the fourth vertex of the rectangle formed by the points (0, 0), (2, 0), (0, 3) are
If P(2, 4), Q(0, 3), R(3, 6) and S(5, y) are the vertices of a parallelogram PQRS, then the value of y is
Write the X-coordinate and Y-coordinate of point P(– 5, 4)
If the vertices of a parallelogram PQRS taken in order are P(3, 4), Q(–2, 3) and R(–3, –2), then the coordinates of its fourth vertex S are ______.
The distance of the point (–6, 8) from x-axis is ______.
The distance of the point (–4, 3) from y-axis is ______.
