General Equation of a Line

If the graphs of the equations $3y – x = 6$ and $3x + 5y = 38$ intersect at a point, then find the coordinates of the point?

The vertices of a triangle $∆OBC$ are $O \left(0, 0\right), B\left(-3, -1\right), C\left(-1, -3\right)$ find the equation of the line parallel to $BC$ and intersecting the sides $OB \& OC$, whose perpendicular distance from the point $\left(0, 0\right)$ is half.

A quadrilateral is formed by the lines $x=0, y=0, x+y=1$ and $6x+y=3$.

What is the equation of other diagonal?

A quadrilateral is formed by the lines $x=0, y=0, x+y=1$ and $6x+y=3$.

What is the equation of diagonal through origin?

Points $P\left(-3,2\right), Q\left(9,10\right) \text{and} R\left(\alpha ,4\right)$ lie on a circle $C$ with $PR$ as its diameter. The tangents to $C$ at the points $Q$ and $R$ intersect at the point $S.$ If $S$ lies on the line $2x-ky=1,$ then $k$ is equal to $?$

The equation of the line passing through the point of intersection of the lines $x+y-5=0,2x-y+4=0$ and having intercepts numerically equal

Consider the triangles with vertices $A\left(2, 1\right), B\left(0, 0\right)$ and $C\left(t, 4\right), t=\left[0, 4\right]$. If the maximum and the minimum perimeters of such triangles are obtained at $t=\alpha$ and $t=\beta$ respectively, then $6\alpha +21\beta$ is equal to ___________.

The straight lines $l_1$ and $l_2$ pass through the origin and trisect the line segment of the line $L:9x+5y=45$ between the axes. If $m_1$ and $m_2$ are the slopes of the lines $l_1$ and $l_2$, then the point of intersection of the line $y=(m_1+m_2)x$ with $L$ lies on

Prove that the area made by $\mathrm{y}={\mathrm{m}}_1\mathrm{x}+{\mathrm{c}}_1, \mathrm{y}={\mathrm{m}}_2\mathrm{x}+{\mathrm{c}}_2 \mathrm{and} \mathrm{x}=0$ is $\frac{1}{2}\frac{{\left({\mathrm{c}}_1-{\mathrm{c}}_2\right)}^2}{\left|{\mathrm{m}}_1-{\mathrm{m}}_2\right|}$.

If the lines $y=3x+1$ and $2y=x+3$ are equally inclined to the line $y=mx+4$, then $m=$

If $\mathrm{P}\left(\mathrm{\theta }\right),\mathrm{Q}(\frac{\mathrm{\pi }}{2}+\mathrm{\theta })$ are tw points on the ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ and $\alpha$ is the angle between the normals $\mathrm{P} \mathrm{and} \mathrm{Q}$ then, 

Let $B\left(1,-3\right)$ and $D\left(0,4\right)$ represent two vertices of rhombus $ABCD$ in $\left(x,y\right)$ plane, then coordinate of vertex $A$ if $\angle BAD=60°$ can be equal to

If the equation of the sides of a triangle are $7x+y-10=0, x-2y+5=0\text{ and} x+y+2 =0$ then find the orthocentre.

The distance of origin from line $\left(1+\sqrt{3}\right)y+\left(1-\sqrt{3}\right)x=10$ measured along the line $y=\sqrt{3}x+k$ is

What is the equation of the line midway between the lines $3x-4y+12=0$ and $3x-4y=6?$

Line $ax+by+p=0$ makes an angle $\frac{\mathrm{\pi }}{4}$ with $x\cos \alpha +y\sin \alpha =p, p\in R^{+}$. If these lines and the line $x\sin \alpha -ycos\alpha =0$ are concurrent, then

Find the equation of straight line parallel to $x+2y-5=0$ and at the same distance from the point $\left(3,2\right)$.

The line $5x+4y=0$ passes through the point of intersection of straight lines