Question

(I) Write down the coordinates of the point P, that divides the line joining $A(-4,1)$and $B(17,10)$ in the ratio $1:2$.

(II) Calculate the distance OP, where O is origin.

(III) Find the equation of a line AB.

Answer 1

Step 1: Finding the coordinates of point $P$ using section formula :

Coordinates of Point $P$ is given by

$P(x,y)=\left(\frac{m_1{x}_2+m_2{x}_1}{m_1+m_2},\frac{m_1{y}_2+m_2{y}_1}{m_1+m_2}\right)$

Here in given question, ${\text{m}}_1=1,m_2=2,x_1=-4,x_2=17,y_1=1,y_2=10$

Coordinates of P is :

$$\begin{gathered} \text{⇒}\left(\frac{1\times 17+2\times (-4)}{1+2},\frac{1\times 10+2\times 1}{1+2}\right) \\ \Rightarrow \left(\frac{17-8}{3},\frac{10+2}{3}\right) \\ \Rightarrow P(x,y)=\left(3,4\right) \end{gathered}$$

Therefore, the coordinates of point $P$ is $(3,4)$

Step 2: Finding the distance $OP$:

Distance formula is given by

$D=$ $\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$

Here we have to find the distance $OP$ from origin$(0,0)$,

substituting , ${\text{x}}_1=0,x_2=3,y_1=0,y_2=4$

$$\begin{gathered} \text{OP =}\sqrt{{(3-0)}^2+{(4-0)}^2} \\ =\sqrt{9+16} \\ =\sqrt{25} \\ =5 \end{gathered}$$

Therefore, the distance $OP$ from origin is $5$.

Step 3: Finding the equation of line $AB$:

The line passing through the points $A(-4,1)$and $B(17,10)$

Equation of Line is given by

$\text{(}y-y_1)=\frac{y_2-y_1}{x_2-x_1}(x-x_1)$

Substitute, $x_1=-4,x_2=17,y_1=1,y_2=10$

$$\begin{gathered} \Rightarrow (\mathrm{y}-\; 1)\; =\left(\frac{10-1}{17\hspace{0.17em}+4}\right)(\mathrm{x}+4) \\ \Rightarrow (\mathrm{y}-1)=\left(\frac{9}{21}\right)(\mathrm{x}+4) \\ \Rightarrow 21(\mathrm{y}-1)=9(\mathrm{x}+\hspace{0.17em}4) \\ \Rightarrow 21\mathrm{y}-21=9\mathrm{x}+\hspace{0.17em}36 \\ \Rightarrow 3\mathrm{x}-7\mathrm{y}+19=0 \end{gathered}$$

Therefore, the coordinates of point $P$ is $(3,4)$, the distance $OP$ from origin is $5$ and the equation of line $AB$ is $3x-\; 7y+\; 19\; =\; 0$.