If the line $6x - 7y + 8 + \lambda (3x - y + 5) = 0$ is parallel to y-axis, then $\lambda$ is equal to
- -7
- -2
- 7
- 2
The Correct Option is A
Solution and Explanation
$6 x-7 y+8+\lambda(3 x-y+5)=0 $
$\Rightarrow (6+3 \lambda) x-(7+\lambda) y+(8+5 \lambda)=0 $
$\Rightarrow (7+\lambda) y=(6+3 \lambda) x+(8+5 \lambda)$
$\Rightarrow y=\frac{3(\lambda+2)}{(\lambda+7)} x+\left(\frac{8+5 \lambda}{7+\lambda}\right)$
$\therefore$ Slope of the line $(m)=\frac{3(\lambda+2)}{(\lambda+7)}$
Since, line is parallel to $y$ -axis.
$m=\infty=\frac{1}{0} $
$\frac{3(\lambda+2)}{\lambda+7}=\frac{1}{0}$
$\Rightarrow \lambda+7=0$
$\Rightarrow \lambda=-7$
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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