If the x-intercept of some line $L$ is double as that of the line, $3x+4y = 12$ and the $y$-intercept of $L$, is half as that of the same line, then the slope of $L$ is:
- $-3$
- $-3/8$
- $-3/2$
- $-3/16$
The Correct Option is D
Solution and Explanation
$\frac{3x}{12}+\frac{4y}{12}=1 \Rightarrow \frac{x}{4}+\frac{y}{3}=1$
$\Rightarrow$ x-intercept=4 and y-intercept= 3 Let the required line be
$L : \frac{x}{a}+\frac{y}{b}=1$ where
$a = x-$intercept and $b=y-$ intercept
According to the question
$a = 4 \times 2 = 8$ and $b = 3/2$
$\therefore$ Required line is $\frac{x}{8}+\frac{2y}{3}=1$
$\Rightarrow 3x+ 16y=24$
$\Rightarrow y=\frac{-3}{16}x+\frac{24}{16}$
Hence, required slope $=\frac{-3}{16}.$
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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
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