In what ratio does the point p 3, 4 divide the line segment joining the points a 1, 2 and b 67

Question 5 Coordinated Geometry - Exercise 7.3

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Answer:

Let the ratio in which x-axis divides the line segment joining (–4, –6) and (–1, 7) = 1: k.

Then,

x-coordinate becomes, \frac{\left(-1-4k\right)}{(k+1)}

y-coordinate becomes, \frac{\left(7-6k\right)}{(k+1)}

Since P lies on x-axis, y coordinate = 0

\frac{\left(7-6k\right)}{(k+1)}=0\\ 7-6k=0\\ k=\frac{7}{6}

Therefore, the point of division divides the line segment in the ratio 6 : 7.

Now, m1 = 6 and m2 = 7

By using the section formula,

x=\frac{\left(m_1x_2+m_2x_2\right)}{(m_1+m_2)}=\frac{\left[6(-1)+7(-4)\right]}{(6+7)}=\frac{\left(-6-28\right)}{13}=-\frac{34}{13}\\ So,\ now\\ y=\frac{\left[6(7)+7(-6)\right]}{(6+7)}=\frac{\left(42-42\right)}{13}=0

Hence, the coordinates of P are (-34/13, 0)

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Let `P(3/4,5/12)` divides AB in the ratio m1:m2

Using the section formula, we have

`(3/4,5/12)=((2m_1+1/2m_2)/(m_1+m_2),(-5m_1+3/2m_2)/(m_1+m_2))`

 `=3/4=(2m_1+1/2m_2)/(m_1+m_2) ....(1)`

 And `5/12=(-5m_1+3/2m_2)/(m_1+m_2)`

 From (1), we have

`3/4=(2m_1+1/2m_2)/(m_1+m_2)`

`=>3(m_1+m_2)=4(2m_1+1/2m_2)`

`=>3m_1+3m_2=8m_1+2m_2`

`=>-5m_1=-m_2`

`therefore m_1/m_2=1/5`

`therefore m_1:m_2=1:5`

Hence, P divides the line segment joining the points A and B internally in the ratio 1 : 5.