Q.2 Â Â Find the coordinates of the points of trisection of the line segment joining (4,â€“1) and (â€“2,â€“3).
Sol. Â Â Â Â Let A(4,â€“1) and B(â€“2,â€“3) be the points of trisection of P and Q.
Â Â Â Â Â Â Â Then, AP = PQ= QB = k (say).
Â Â Â Â Â Â Â Therefore, PB = PQ + QB = 2k
Â Â Â Â Â Â Â and, AQ = AP + PQ = 2k
Â Â Â Â Â Â Â AP : PB = k : 2k = 1 : 2
Â Â Â Â Â Â Â and AQ : QB = 2k : k = 2 : 1
Â Â Â Â Â Â Â So, P divides AB internally in the ratio 1 : 2, while Q divides AB internally in the ratio 2 : 1.
Â Â Â Â Â Â Â Thus, the coordinates of
Â Â Â Â Â Â P areÂ
Â Â Â Â Â Â Â and, the coordinates of Q areÂ Â
Â Â Â Â Â Â Â Hence, the two points of trisection are and .
Q.3 Â Â To conduct Sports Day activities in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1 m each. 100 flowerÂ pots have been placed at a distance of 1 m from each other along AD, as shown in figure.Â Niharika runs th the distance AD on the 2nd line and posts a green flag. Preet runs th Â the distance AD on the eighth line and posts a red flag. What is theÂ distance betweenÂ both the flags ? If Rashmi has to post a blue flag exactly halfwayÂ between the lineÂ (segment) joining the two flags, where should she post her flag?
Sol. Â Â Clearly from the figure, the position of green flag posted by Niharika is given by ,i.e., P(2, 25)
Â Â Â Â Â Â and that of red flag posted by Preet is given by Â i.e., Q(8, 20).
Â Â Â Â Â Â Now,
Â Â Â Â Â Â Â Â Â Â Â Â Â Â
Â Â Â Â Â Â Â Â Â Â Â Â Â Â
Â Â Â Â Â Â Therefore, the distance between the flags = metres
Â Â Â Â Â Â Let M be the position of the blue flag posted by Rashmi in the halfway of line segment PQ.
Â Â Â Â Â Therefore, M is the given by Â or ,i.e., (5, 22.5) .
Â Â Â Â Â Thus, the blue flag is on the fifth line at a distance 22.5m above it.
Q.4 Â Â Â Find the ratio in which the line segment joining the points of (â€“3, 10) and (6, â€“ 8) is divided byÂ (â€“1, 6).
Sol.Â Â Â Â Let the point P(â€“1, 6) divide the line joining A(â€“3, 10) and B(6, â€“8) in the ratio k : 1. Then, the
Â Â Â Â Â Â Â coordinates of P areÂ .Â
Â Â Â Â Â Â Â But, the coordinates of P are given as (â€“1, 6).
Â Â Â Â Â Â Â Therefore, and
Â Â Â Â Â Â Â and
Â Â Â Â Â Â Â and
Â Â Â Â Â Â Â and
Â Â Â Â Â Â Â
Â Â Â Â Â Â Â Hence, the point P divides AB in the ratio 2 : 7.
Q.5 Â Â Â Find the ratio in which the line segment joining A(1, â€“5) and B(â€“ 4, 5) is divided by theÂ x-axis. Also find the coordinates of the point of division.
Sol.Â Â Â Â Let the required ratio be k : 1. Then, the coordinates of the point P of division are .
Â Â Â Â Â Â Â But it is a point on x-axis on which y - coordinate of every point is zero.
Â Â Â Â Â Â Â Therefore,
Â Â Â Â Â Â Â
Â Â Â Â Â Â Â 5k = 5
Â Â Â Â Â Â Â k = 1
Â Â Â Â Â Â Thus, the required ratio is 1 : 1 and the point of division P is given by
Â Â Â Â Â Â ,i.e., i.e.,
Â Â Â Â Â Â Â Therefore,
Â Â Â Â Â Â Â x + 1 = 7
Â Â Â Â Â Â Â x = 6
Â Â Â Â Â Â Â and,
Â Â Â Â Â Â Â 5 + y = 8
Â Â Â Â Â Â Â y = 3
Â Â Â Â Â Â Â Hence, x = 6 and y = 3
Q.7 Â Â Â Find the coordinates of a point A, where AB is the diameter of a circle whose centre isÂ (2, â€“3) andÂ B is (1, 4).
Sol.Â Â Â Â Let AB be a diameter of the circle having its centre at C(2, â€“3) such that the coordinates of end B are (1, 4).
Â Â Â Â Â Â Â Let the coordinates of A be (x, y).
Â Â Â Â Â Â Â Since C is the mid - point of AB , therefore
Â Â Â Â Â Â Â
Â Â Â Â Â Â Â and,
Â Â Â Â Â Â Â Hence, the coordinates of A are (3, â€“10) .
Q.8 Â Â Â If A and B are (â€“2, â€“2) and (2, â€“ 4) respectively, find the coordinates of P such thatÂ AP = AB and P liesÂ on the line segment AB.
Sol.Â Â Â Â We have,
Â Â Â Â Â Â
Â Â Â Â Â Â
Â Â Â Â Â Â Â
Â Â Â Â Â Â
Â Â Â Â Â Â Let P(x, y) be the point which divides the join of A(â€“2, â€“ 2) and B(2, â€“ 4) in the ratio 3 : 4.
Â Â Â Â Â Â Therefore, Â
Â Â Â Â Â Â and,
Â Â Â Â Â Â Hence, the coordinates of the point P are .
Q.9 Â Â Find the coordinates of the points which divide the line segment joining A(â€“ 2, 2) andÂ B(2, 8) into four equal parts.
Sol. Â Â Â Let and be the points that divide the line segment joining A(â€“2, 2) and B(2, 8) intoÂ fourÂ equal parts.
Â Â Â Â Â Â Since divides the segment into two equal parts
Â Â Â Â Â Therefore, coordinates of (i.e., mid - point) are
Â Â Â Â Â , i.e., (0, 5).
Â Â Â Â Â Now, divides the line segment into two equal parts.
Â Â Â Â Â Therefore, coordinates of (i.e., mid - point) are
Â Â Â Â Â , i.e., .
Â Â Â Â Â Â Â Again,Â is the mid point of line segment .Â
Â Â Â Â Â Therefore, coordinates of are , i.e., .
Q.10 Â Â Find the area of a rhombus if its vertices are (3, 0), (4, 5), (â€“ 1, 4) andÂ (â€“ 2,â€“1) taken in order.
Sol.Â Â Â Â Let A(3, 0), B(4, 5), C(â€“1, 4) and D(â€“2,â€“ 1) be the vertices of the rhombus ABCD.
Â Â Â Â Â Â Â Diagonal Â Â Â and, diagonal
Â Â Â Â Â
Â Â Â Â Â Â Â Area of the rhombus ABCD
Â Â Â Â Â Â Â (Product of lengths of diagonals )
Â Â Â Â Â Â Â
Â Â Â Â Â Â Â sq. units
Â Â Â Â Â Â Â = 24 sq. units
Â
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