The line passing through the intersection of | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The family of lines passing through the intersection of $ax + 2by + 3b = 0$ and $bx - 2ay - 3a = 0$ is given by $(ax + 2by + 3b) + \lambda(bx - 2a

Vedclass · Accepted Answer

At a distance of $3/2$ below the $x$-axis

Vedclass · Answer

(A) The family of lines passing through the intersection of $ax + 2by + 3b = 0$ and $bx - 2ay - 3a = 0$ is given by $(ax + 2by + 3b) + \lambda(bx - 2ay - 3a) = 0$.Rearranging the terms,we get $(a + \lambda b)x + (2b - 2a\lambda)y + (3b - 3a\lambda) = 0$.Since the line is parallel to the $x$-axis,the coefficient of $x$ must be zero.Therefore,$a + \lambda b = 0$,which gives $\lambda = -a/b$.Substituting $\lambda = -a/b$ into the equation:$(ax + 2by + 3b) - \frac{a}{b}(bx - 2ay - 3a) = 0$$ax + 2by + 3b - ax + \frac{2a^2}{b}y + \frac{3a^2}{b} = 0$$y(2b + \frac{2a^2}{b}) + (3b + \frac{3a^2}{b}) = 0$$y(\frac{2b^2 + 2a^2}{b}) = -(\frac{3b^2 + 3a^2}{b})$$y = -\frac{3(a^2 + b^2)}{2(a^2 + b^2)} = -\frac{3}{2}$.Thus,the line is $y = -3/2$,which is at a distance of $3/2$ below the $x$-axis.

The line passing through the intersection of the lines $ax + 2by + 3b = 0$ and $bx - 2ay - 3a = 0$ (where $(a, b) \neq (0, 0)$) and parallel to the $x$-axis is:

Similar Questions

The lines $(a+2b)x + (a-3b)y = a-b$ for different values of $a$ and $b$ pass through a fixed point whose coordinates are:

The equation of the straight line perpendicular to the straight line $3x + 2y = 0$ and passing through the point of intersection of the lines $x + 3y - 1 = 0$ and $x - 2y + 4 = 0$ is

If the lines $ax + by + c = 0$,$bx + cy + a = 0$,and $cx + ay + b = 0$ are concurrent,then:

The line parallel to the $x$-axis passing through the intersection of the lines $ax + 2by + 3b = 0$ and $bx - 2ay - 3a = 0$, where $(a, b) \neq (0, 0)$, is

If $x+2y-3=0$,$3x+4y-7=0$,$2x+3y-4=0$,and $4x+5y-6=0$ are the equations of four lines,then

Vedclass Test Series

Exam Paper Generator

Online Exam Module