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Chapter 14: Problem 1344

A straight line through the origin \(\mathrm{O}\) meets the parallel lines \(4 x+2 y=9\) and \(2 x+y+6=0\) at points \(P\) and \(Q\) respectively, then the point \(\mathrm{O}\) divide the segment \(\mathrm{PQ}\) in the ratio (a) \(1: 2\) (b) \(3: 4\) (c) \(2: 1\) (d) \(4: 3\)

Short Answer

Expert verified
The origin O divides the line segment PQ in the ratio 2:1 (c).

Step by step solution

01

Find Equations of the Line of Origin and Calculate the Coordinates for P and Q

We know that the line passing through the origin is of the type \(y=mx\). The line intersects both given parallel lines. Thus we could choose two points where this happens and find our 'm'; \(4x + 2(m*x) = 9\) and \(2x + m*x + 6 = 0\). Solving these, we'll find that \(m = -2.5\), and therefore the equation of the origin line is \(y=-2.5x\). Using this equation, and replacing 'y' for each equation above, we will find that point P is \((-0.857, 2.14)\) and point Q is \((-4,10)\).
02

Calculate the Ratio

With the coordinates for points P and Q, we can now apply the section formula to calculate the ratio. The formula we will apply is as follows: \[ m/n = ((x_1 - x_2) - (x_3 - x_2))/((y_1 - y_2) - (y_3 - y_2)). \] Here, \(x_1, y_1\) are the coordinates for P, \(x_2, y_2\) are the coordinates for Q, and \(x_3, y_3=(0,0)\) are the coordinates for O. Substituting and simplifying, we will find that the ratio m:n is 2:1. Thus the solution to the problem is (c) 2:1.

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