In what ration does the point (24/11,Y) Divided the line segment joining the point p(2,-2) and q(3,7)? Also find the value of y?

Question

Answer

Let the point A(24/11, y) divides the segment PQ, given coordinates are P(2,-2) & Q(3,7) .

Here,
x1 = 2,x2 = 3,

y1= -2,y2 = 7

By section formula,

X coordinate of point A = (m1 X2 + m2 X1 ) / (m1 + m2)

Here, let the ratio m1: m2 be k:1

So, 24/11 = (3k + 2) / ( k+1)

=> 33k +22 = 24k +24

=> 9k = 2

=> k = 2/9

So, the ratio = 2:9

That's all🙂

Answer

Answer

Introduction:
We are given the points P(2, -2) and Q(3, 7) in a coordinate plane. Our task is to find the ratio in which the point (24/11, Y) divides the line segment joining P and Q, and also determine the value of Y.

Procedure:
To find the ratio in which the point (24/11, Y) divides the line segment PQ, we need to calculate the distances between point P and (24/11, Y) as well as between (24/11, Y) and point Q.

Calculating the Distance:
The distance formula is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

1. Calculating the distance between point P and (24/11, Y):
- x1 = 2, y1 = -2
- x2 = 24/11, y2 = Y
- d1 = sqrt((24/11 - 2)^2 + (Y - (-2))^2)
= sqrt((24/11 - 22/11)^2 + (Y + 2)^2)
= sqrt((2/11)^2 + (Y + 2)^2)
= sqrt(4/121 + (Y + 2)^2)
= sqrt((4 + (Y + 2)^2)/121)

2. Calculating the distance between (24/11, Y) and point Q:
- x1 = 24/11, y1 = Y
- x2 = 3, y2 = 7
- d2 = sqrt((3 - 24/11)^2 + (7 - Y)^2)
= sqrt((33/11 - 24/11)^2 + (7 - Y)^2)
= sqrt((9/11)^2 + (7 - Y)^2)
= sqrt(81/121 + (7 - Y)^2)
= sqrt((81 + 121(7 - Y)^2)/121)

Ratio Calculation:
The ratio in which the point (24/11, Y) divides the line segment PQ is given by:
ratio = d1/d2

Substituting the values of d1 and d2:
ratio = sqrt((4 + (Y + 2)^2)/121) / sqrt((81 + 121(7 - Y)^2)/121)
= sqrt((4 + (Y + 2)^2) / (81 + 121(7 - Y)^2))

Solving for Y:
To find the value of Y, we can equate the ratio to a given value and solve for Y algebraically. For example, if the ratio is 1:2, we can set up the equation:

sqrt((4 + (Y + 2)^2) / (81 + 121(7 - Y)^2)) = 1/2

Squaring both sides of the equation and simplifying, we can solve for Y. However, since the specific ratio is not given in the question, we cannot determine the exact value of Y without additional information.

Conclusion:
To summarize, we have calculated the ratio in which the point (24/

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