Solveeit Logo

Question

Question: A straight line through origin O meets the parallel lines \(4x+2y=9\) and \(2x+y+6=0\) at point P an...

A straight line through origin O meets the parallel lines 4x+2y=94x+2y=9 and 2x+y+6=02x+y+6=0 at point P and Q respectively. Then the point O divides the segment PQ in the ratio.
a)1:2 b)3:4 c)2:1 d)4:3 \begin{aligned} & a)1:2 \\\ & b)3:4 \\\ & c)2:1 \\\ & d)4:3 \\\ \end{aligned}

Explanation

Solution

Now we are given that a straight line through origin O meets the parallel lines 4x+2y=94x+2y=9 and 2x+y+6=02x+y+6=0 at point P and Q respectively. We know that equation of line passing through origin is in y=mxy=mx hence we will use this equation to solve with both lines to get the coordinates P and Q. Now we have coordinates of P, Q and O. hence we can use section formula which says if (x, y) divides the line joining (x1,x2)\left( {{x}_{1}},{{x}_{2}} \right) and (y1,y2)\left( {{y}_{1}},{{y}_{2}} \right) in ration m : n. then we have.
(x,y)=((mx1+nx2m+n),(my1+ny2m+n))\left( x,y \right)=\left( \left( \dfrac{m{{x}_{1}}+n{{x}_{2}}}{m+n} \right),\left( \dfrac{m{{y}_{1}}+n{{y}_{2}}}{m+n} \right) \right) . hence we can find the ratio m : n.

Complete step-by-step answer :

Now consider the equation of line passing through origin
We know that the equation of line passing through origin is y=mxy=mx
Now let us say the equation of line PQ is y=mxy=mx
Now Let us first find the point of intersection P.
P lies on line 4x+2y=94x+2y=9 and the line y=mxy=mx
Now consider the line 4x+2y=94x+2y=9
Dividing throughout by 2 we get
2x+y=922x+y=\dfrac{9}{2}
Hence we get y=922xy=\dfrac{9}{2}-2x
Now at intersection point of line 4x+2y=94x+2y=9 and the line y=mxy=mx which is P we have

& mx=\dfrac{9}{2}-2x \\\ & mx+2x=\dfrac{9}{2} \\\ & x\left( m+2 \right)=\dfrac{9}{2} \\\ & x=\dfrac{9}{2\left( m+2 \right)} \end{aligned}$$ Now substituting $$x=\dfrac{9}{2\left( m+2 \right)}$$ in $y=mx$ we get. $y=\dfrac{9m}{2\left( m+2 \right)}$ Hence the coordinates of P is $\left( \dfrac{9}{2\left( m+2 \right)},\left( \dfrac{9m}{2\left( m+2 \right)} \right) \right).............(1)$ Now line $2x+y+6=0$ and $y=mx$ intersects at Q. Rearranging the terms of $2x+y+6=0$ we get $y=-2x-6$ Now at interaction point Q of line $2x+y+6=0$ and $y=mx$we will have $mx=-2x-6$ Hence we get $\begin{aligned} & mx+2x=6 \\\ & \Rightarrow x=\dfrac{6}{\left( m+2 \right)} \\\ \end{aligned}$ Now substituting $x=\dfrac{6}{\left( m+2 \right)}$ in $y=mx$ we get $y=\dfrac{6m}{\left( m+2 \right)}$ Hence the coordinates of Q are $\left( \dfrac{6}{m+2},\dfrac{6m}{m+2} \right).....................(2)$ Now we have coordinates of P is $\left( \dfrac{9}{2\left( m+2 \right)},\left( \dfrac{9m}{2\left( m+2 \right)} \right) \right)$ and coordinates of Q are $\left( \dfrac{6}{m+2},\dfrac{6m}{m+2} \right)$ Now we have O = (0, 0) divides the line PQ internally. Let us say that that the point O divides the line PQ in ratio λ : 1. Then we know by section formula if (x, y) divides the line joining $\left( {{x}_{1}},{{x}_{2}} \right)$ and $\left( {{y}_{1}},{{y}_{2}} \right)$ in ration m : n. then we have. $\left( x,y \right)=\left( \left( \dfrac{m{{x}_{1}}+n{{x}_{2}}}{m+n} \right),\left( \dfrac{m{{y}_{1}}+n{{y}_{2}}}{m+n} \right) \right)$ Hence for the line PQ we have. $\left( 0,0 \right)=\left( \dfrac{\left( \dfrac{\lambda 9}{2\left( m+2 \right)}-\dfrac{6}{m+2} \right)}{\lambda +1},\dfrac{\left( \dfrac{\lambda 9m}{2\left( m+2 \right)}-\dfrac{6m}{m+2} \right)}{\lambda +1} \right)$ Now first equating x coordinate we get $$\begin{aligned} & \dfrac{\left( \dfrac{\lambda 9}{2\left( m+2 \right)}-\dfrac{6}{m+2} \right)}{\lambda +1}=0 \\\ & \Rightarrow \left( \dfrac{\lambda 9}{2\left( m+2 \right)}-\dfrac{6}{m+2} \right)=0 \\\ & \Rightarrow \dfrac{\lambda 9}{2\left( m+2 \right)}=\dfrac{6}{m+2} \\\ & \Rightarrow \dfrac{3\lambda }{2}=\dfrac{2}{1} \\\ & \Rightarrow \lambda =\dfrac{4}{3} \\\ \end{aligned}$$ Hence the value of λ is $\dfrac{4}{3}$ . Now we have point O divides the line PQ in ratio λ : 1. Hence O divides PQ in $\dfrac{4}{3}:1=4:3$ Hence we have O divides line PQ in 4 : 3. **Option d is the correct answer.** **Note** : In section formula we use the ratio as m : n, and to solve we have assumed the ratio to be λ : 1 for simplicity. The answer through both the methods will be the same.