Find the ratio in which point P(k, 7) divides the segment joining A(8, 9) and B(1, 2). Also find k .
Find the ratio in which point P(k, 7) divides the segment joining A(8, 9) and B(1, 2). Also find k .
Answer:-
Let the ratio be x : 1.
Using the section formula,
[k = frac{1x + 1 times 8}{x + 1}]
[ Rightarrow kx + k = x + 8 . . . . . left( 1 right)]
[Also, 7 = frac{2x + 9}{x + 1}]
[ Rightarrow 7x + 7 = 2x + 9]
[ Rightarrow 5x = 2]
[ Rightarrow x = frac{2}{5}]
So, the required ratio is 2 : 5.
Putting this value of x in (1) we get
[kleft( frac{2}{5} + 1 right) = frac{2}{5} + 8]
[ Rightarrow frac{7}{5}k = frac{42}{5}]
[ Rightarrow k = 6]
Answer:_
Let’s first find the coordinates of point P.
The x-coordinate of P can be found using the formula for the partition of a line segment by a point:
x-coordinate of P = [(x-coordinate of B – x-coordinate of A) * k + x-coordinate of A * (7 – 2)] / (7 – 2) = [(1 – 8) * k + 8 * 5] / 5 = (-7k + 40) / 5
Similarly, the y-coordinate of P can be found using the same formula:
y-coordinate of P = [(y-coordinate of B – y-coordinate of A) * k + y-coordinate of A * (7 – 2)] / (7 – 2) = [(2 – 9) * k + 9 * 5] / 5 = (-7k + 45) / 5
So the coordinates of point P are:
P((-7k + 40) / 5, (-7k + 45) / 5)
To find the ratio in which P divides the line segment AB, we can use the distance formula:
AP = sqrt((k – 8)^2 + (7 – 9)^2) = sqrt((k – 8)^2 + 4) BP = sqrt((1 – k)^2 + (2 – 7)^2) = sqrt((1 – k)^2 + 25) AB = sqrt((8 – 1)^2 + (9 – 2)^2) = sqrt(170)
By the segment partition formula, we have:
AP/PB = BP/AB
(sqrt((k – 8)^2 + 4)) / (sqrt((1 – k)^2 + 25)) = (sqrt((1 – k)^2 + 25)) / sqrt(170)
Squaring both sides and simplifying, we get:
(k – 8)^2 + 4 = 170 * (1 – k)^2 + 25 * (1 – k)
Expanding and simplifying, we get:
168k^2 – 556k + 449 = 0
Solving this quadratic equation using the quadratic formula, we get:
k = (139 + sqrt(5921)) / 84 or k = (139 – sqrt(5921)) / 84
Substituting either value of k back into the equation for the coordinates of P, we get the coordinates of P. We can then use the distance formula to check that AP/PB = BP/AB.
Chapter 5. Co-ordinate Geometry – Practice Set 5.2 (Page 115)