A straight line through the origin O meets the parallel lines $4x + 2y = 9 $ and $2x + y + 6 = 0$ at points P and Q respectively. Then, the point O divides the segment PQ in the ratio
1:2
3:4
2:1
4:3
The Correct Option is B
Solution and Explanation
Given that;
The equation of the line is:-
\(4x+2y=9-(i)\) and \(2x+y=-6-(ii)\)
Let the equation of the line passes through the origin: \(y=mx-(iii)\)
Then two intersection of (i) and (iii)
\(x_1=\frac{9}{4+2m},y=\frac{9m}{4+2m}|p(\frac{9}{4+2m},\frac{9m}{4+2m})\)
similarly, (ii) and (iii);
\(x_2=\frac{-6}{2+m},y_2=\frac{-6m}{2+m}|q(\frac{-6}{2+m},\frac{-6m}{2+m})\)
Let us consider a point ‘O’ denotes PQ in the ratio of \(1\ratio{K}\)
\(\therefore 0=\frac{K(\frac{9}{4+2m}-\frac{6}{2+m})}{K+m}\) and \(\frac{K(\frac{9m}{4+2m}-\frac{6m}{2+1})}{K+1}\)
\(K=\frac{6\times2}{9}=\frac{4}{3}\)
\(\therefore K=\frac{4}{3}\) \(\therefore Ratio\space is \space3\ratio{4}\)
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Concepts Used:
Straight lines
A straight line is a line having the shortest distance between two points.
A straight line can be represented as an equation in various forms, as show in the image below:
The following are the many forms of the equation of the line that are presented in straight line-
1. Slope – Point Form
Assume P0(x0, y0) is a fixed point on a non-vertical line L with m as its slope. If P (x, y) is an arbitrary point on L, then the point (x, y) lies on the line with slope m through the fixed point (x0, y0) if and only if its coordinates fulfil the equation below.
y – y0 = m (x – x0)
2. Two – Point Form
Let's look at the line. L crosses between two places. P1(x1, y1) and P2(x2, y2) are general points on L, while P (x, y) is a general point on L. As a result, the three points P1, P2, and P are collinear, and it becomes
The slope of P2P = The slope of P1P2 , i.e.
\(\frac{y-y_1}{x-x_1} = \frac{y_2-y_1}{x_2-x_1}\)
Hence, the equation becomes:
y - y1 =\( \frac{y_2-y_1}{x_2-x_1} (x-x1)\)
3. Slope-Intercept Form
Assume that a line L with slope m intersects the y-axis at a distance c from the origin, and that the distance c is referred to as the line L's y-intercept. As a result, the coordinates of the spot on the y-axis where the line intersects are (0, c). As a result, the slope of the line L is m, and it passes through a fixed point (0, c). The equation of the line L thus obtained from the slope – point form is given by
y – c =m( x - 0 )
As a result, the point (x, y) on the line with slope m and y-intercept c lies on the line, if and only if
y = m x +c