Question

Find the ratio in which the point P(2.x) divides the line joining the point
A(-2, 2) and B(3,7) internally. Also find the value of x.​

Answer 1

AnswEr:-

Value of x is 6.

Step by Step Explanation:-

Let us consider that point P(2 , x) divides the line segment joining the points A(-2, 2) and B(3, 7) in the ratio of k:1.

By using Section Formula:-

$:\implies\sf\; \dfrac{m_1\; x_2 + m_2\; x_1}{m_1 \; + \; m_2}\; ,\, \dfrac{m_1\;y_2+m_2\;y_1}{m_1+m_2}$

$\rule{150}3$

$:\implies\sf\;(2,x) = \dfrac{(k)(3) + (1) (-2)}{k + 1}\; ,\; \dfrac{(k)(7) + (1)(2)}{k + 1}$

$:\implies\sf\;(2 , x) = \dfrac{ 3k - 2}{k + 1}\; ,\; \dfrac{7 k + 2}{k + 1}$

$:\implies\sf\;2 = \dfrac{3k - 2}{k + 1}$

$:\implies\sf\; 2 (k + 1) = 3k - 2$

$:\implies\sf\;2k - 3k = -2 - 2$

$:\implies\sf\; - k = - 4$

$:\implies\large\boxed{\sf{\pink{ k = 4}}}$

$\rule{150}2$

The Point (2 ,x) divides the line segment in the ratio of 4:1.

$:\implies\sf\;x = \dfrac{7k + 2}{k + 1}$

Putting the value of k

$:\implies\sf\; x =\dfrac{7(4) + 2}{4 + 1}$

$:\implies\sf\; x = \dfrac{30}{5}$

$:\implies\large\boxed{\sf{\pink{x = 6}}}$

$\rule{150}3$

Answer 2

Answer:

Given :

The coordinates of A (-2,2), B(3,7).

Coordinate of the point P (2,x).

To find :

The ratio in which the line AB is divided at point P =?

x =?

Formula used :

Section formula,

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

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

Here m, n is the ratio and (x, y) are the required coordinates.

Then according to the formula let (x1, y1) & (x2, y2) be A(-2,2) & B(3,7),

Then,

$\frac{3m - 2n}{m + n} = 2...........eq1$

And,

$\frac{7m + 2n}{m + n} = x...........eq2$

From eq1 we get,

$= > 3m - 2n = 2m + 2n \\ = > 3m - 2m = 2n + 2n \\ = > m = 4n \\ = > \frac{m}{n} = \frac{4}{1}$

So m:n = 4:1

Putting this value in equation 2 we get,

$= > \frac{7m + 2n}{m + n} = x \\ = > \frac{7 \times 4 + 2 \times 1}{4 + 1} = x \\ = > \frac{30}{5} = x = 6$

Therefore x = 6.

The coordinates of P are (2,6).

The coordinates of P are (2,6).The ratio m:n = 4:1.