Question

in what ratio does the point ( 24/11,y) divide the line segment joining the points P(2.-2) and Q(3,7) ? also find the value of Y

Answer 1

It will divided into 2:9.
y coordinate =-4/11

Answer 2

Given:

A point (24/11, y) divides the line segment joining the points P(2,-2) and Q(3,7).

To find:

The value of Y and the ratio in which the given point (24/11, y) divides the line segment joining the points P(2,-2) and Q(3,7).

Solution:

If a point P (x, y) divides the line segment AB joining the points A(x1, y1) and B(x2, y2) in the ratio m:n, then the coordinates of the point is given by:

$x = \frac{mx2 \: + \: nx1}{m \: + \: n}$

$y = \frac{my2 \: + \: ny1}{m \: + \: n}$

So, let the point (24/11, y) divide the line segment joining the points P(2,-2) and Q(3,7) in the ratio k:1. So, we have,

$\frac{24}{11} = \frac{3(k) \: + \: 2(1)}{k \: + \: 1} \: \: (i)$

and,

$y = \frac{7(k) + ( - 2)(1)}{k \: + \: 1} \: \: (ii)$

On solving (i), we have,

$24k \: + 24 = 33k + 22$

$9k = 2$

$k = \frac{2}{9}$

As the point divides the given line segment PQ in the ratio k:1 and k = 2/9, so, point divides the line segment joining the points P(2.-2) and Q(3,7) in the ratio 2:9.

Now,

On putting the value of k in (ii), we have,

$y = \frac{7( \frac{2}{9} ) + ( - 2)(1)}{ \frac{2}{9} \: + \: 1}$

On solving and taking 9 as LCM, we have

$y = \frac{14 - 18}{11}$

$y = \frac{ - 4}{11}$

Hence, the ratio in which the given point (24/11, y) divides the line segment joining the points P(2,-2) and Q(3,7) is 2:9. Also, the value of y is -4/11.