Let $a ,b,c$ and $d$ be non-zero numbers. If the point of intersection of the lines $4ax+2ay+c = 0$ and $5bx + 2by+ d = 0$ lies in the fourth quadrant and is equidistant from the two axes, then
- $2bc - 3ad = 0$
- $2bc + 3ad = 0$
- $ 3bc - 2ad= 0$
- $3bc + 2ad = 0$
The Correct Option is C
Solution and Explanation
$\therefore 4 a \alpha-2 a \alpha+ c =0$
$ \Rightarrow \alpha=-\frac{c}{2a}$
and $5b \alpha-2 b \alpha+ d =0$
$ \Rightarrow \alpha=-\frac{d}{3 b}$
$\Rightarrow 3 b c=2 a d $
$\Rightarrow 3 b c-2 a d=0$
:
The point of intersection will be
$ \frac{x}{2 a d-2 b c}=\frac{-y}{4 a d-5 b c}=\frac{1}{8 a b-10 a b} $
$\Rightarrow x=\frac{2(a d-b c)}{-2 a b} $
$\Rightarrow y=\frac{5 b c-4 a d}{-2 a b}$
$\because$ Point of intersection is in fourth quadrant so x is positive and y is negative
Also distance from axes is same
So $x = - y $ $(\because$ distance from x-axis is - y as y is negative)
$\frac{2(a d-b c)}{-2 a b}=\frac{-(5 b c-4 a d)}{-2 a b}$
$2 a d-2 b c=-5 b c+4 a d$
$\Rightarrow 3 b c-2 a d=0\,\,\,\,\,\,\dots(i)$
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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