JEE Main · 2023 · 15 Apr 2023 · Shift 1 · Straight Lines

Let a point A lie between the parallel lines ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ such that its distances from ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle ABC , where the points B and C l …

Among the statements
(S1): If $A(5,-1)$ and $B(-2,3)$ are two vertices of a triangle, whose orthocentre is $(0,0)$, then its third vertex is $(-4,-7)$ and
(S2): If positive numbers $2a,b,c$ are three consecutive terms of an A.P., then the lines $ax+by+c=0$ are …

A rectangle is formed by the lines $\mathrm{x}=0$, $\mathrm{y}=0$, $\mathrm{x}=3$ and $\mathrm{y}=4$. Let the line L be perpendicular to $3x+y+6=0$ and divide the area of the rectangle into two equal parts. Then the distance of the point $\left(\frac{1}{2},-5\right)$ …

Let $\mathrm{A}(1,2)$ and $\mathrm{C}(-3,-6)$ be two diagonally opposite vertices of a rhombus, whose sides AD and BC are parallel to the line $7\mathrm{x}-\mathrm{y}=14$. If $\mathrm{B}(\alpha ,\beta )$ and $\mathrm{D}(\gamma ,\delta )$ are the other two vertices, then …

Let $A(1,0),B(2,-1)$ and $C\left(\frac{7}{3},\frac{4}{3}\right)$ be three points. If the equation of the bisector of the angle ABC is $\alpha x+\beta y=5$, then the value of ${\alpha }^2+{\beta }^2$ is

Let the angles made with the positive x-axis by two straight lines drawn from the point $\mathrm{P}(2,3)$ and meeting the line $\mathrm{x}+\mathrm{y}=6$ at a distance $\sqrt{\frac{2}{3}}$ from the point P be ${\theta }_1$ and ${\theta }_2$. Then the value of …