The equation of the line passing through the | vedclass

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

Question

(A) The equation of the family of lines passing through the intersection of $3x - 4y + 1 = 0$ and $5x + y - 1 = 0$ is given by $(3x - 4y + 1) + \lambd

Vedclass · Accepted Answer

$23x + 23y - 11 = 0$

Vedclass · Answer

(A) The equation of the family of lines passing through the intersection of $3x - 4y + 1 = 0$ and $5x + y - 1 = 0$ is given by $(3x - 4y + 1) + \lambda(5x + y - 1) = 0$. Rearranging the terms,we get $(3 + 5\lambda)x + (\lambda - 4)y + (1 - \lambda) = 0$. For the line to cut off equal intercepts on the axes,the coefficients of $x$ and $y$ must be equal in magnitude or the line must pass through the origin. If the intercepts are equal,then the slope of the line must be $-1$ (for intercepts $a, a$) or $1$ (for intercepts $a, -a$). Case $1$: Slope $m = -\frac{3 + 5\lambda}{\lambda - 4} = -1$ $\Rightarrow 3 + 5\lambda = \lambda - 4$ $\Rightarrow 4\lambda = -7$ $\Rightarrow \lambda = -\frac{7}{4}$. Substituting $\lambda = -\frac{7}{4}$ into the equation: $(3 - \frac{35}{4})x + (-\frac{7}{4} - 4)y + (1 + \frac{7}{4}) = 0$ $\Rightarrow -23x - 23y + 11 = 0$ $\Rightarrow 23x + 23y - 11 = 0$. Case $2$: The line passes through the origin $(1 - \lambda = 0 \Rightarrow \lambda = 1)$,which gives $8x - 3y = 0$ (intercepts are $0, 0$,which are equal). Since $23x + 23y - 11 = 0$ is one of the options,option $A$ is correct.

The equation of the line passing through the intersection of $3x - 4y + 1 = 0$ and $5x + y - 1 = 0$ which cuts off equal intercepts on the axes is given by

Similar Questions

One side of a rectangle lies along the line $4x + 7y + 5 = 0.$ Two of its vertices are $(-3, 1)$ and $(1, 1).$ Then the equations of the other three sides are

$A$ square of side $a$ lies above the $x$-axis and has one vertex at the origin. The side passing through the origin makes an angle $\alpha, (0 < \alpha < \frac{\pi}{4})$ with the positive direction of the $x$-axis. The equation of its diagonal not passing through the origin is

Let a variable line of slope $m > 0$ passing through the point $(4, -9)$ intersect the coordinate axes at the points $A$ and $B$. The minimum value of the sum of the distances of $A$ and $B$ from the origin is

The number of integer values of $m$,for which the $x$-coordinate of the point of intersection of the lines $3x + 4y = 9$ and $y = mx + 1$ is also an integer,is

The number of possible distinct straight lines passing through $(2,3)$ and forming a triangle with the coordinate axes whose area is $12$ sq. units is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module