Embibe Experts Solutions for Chapter: Point and Straight Line, Exercise 1: JEE Main - 1 February 2023 Shift 1

Let $A\left(\alpha ,-2\right),B\left(\alpha ,6\right)$ and $C\left(\frac{\alpha }{4},-2\right)$ be vertices of a $∆ABC$. If $\left(5,\frac{\alpha }{4}\right)$ is the circumcentre of $∆ABC$, then which of the following is NOT correct about $∆ABC$

The equations of the sides $AB, BC \& CA$ of a triangle $ABC$ are $2x+y=0$, $x+py=21a$ $\left(a\ne 0\right)$ and $x-y=3$ respectively. Let $P\left(2,a\right)$ be the centroid of the triangle $ABC$, then ${\left(BC\right)}^2$ is equal to

A triangle is formed by $X$-axis, $Y$-axis and the line $3x+4y=60$. Then the number of points $P\left(a,b\right)$ which lie strictly inside the triangle, where $a$ is an integer and $b$ is a multiple of $a$, is _____ .

A light ray emits from the origin making an angle $30°$ with the positive $x-$axis. After getting reflected by the line $x+y=1$, if this ray intersects $x-$axis at $Q$, then the abscissa of $Q$ is

Let $B$ and $C$ be the two points on the line $y+x=0$ such that $B$ and $C$ are symmetric with respect to the origin. Suppose $A$ is a point on $\mathrm{y}-2\mathrm{x}=2$ such that $∆ABC$ is an equilateral triangle. Then, the area of the $∆ABC$ is

A straight line cuts off the intercepts $\mathrm{OA}=\mathrm{a}$ and $\mathrm{OB}=\mathrm{b}$ on the positive directions of $\mathrm{x}$-axis and $\mathrm{y}-$ axis respectively. If the perpendicular from origin $\mathrm{O}$ to this line makes an angle of $\frac{\pi}{6}$ with positive direction of $y$-axis and the area of $\triangle \mathrm{OAB}$ is $\frac{98}{3} \sqrt{3}$, then $\mathrm{a}^2-\mathrm{b}^2$ is equal to:

If the orthocentre of the triangle, whose vertices are $\left(1,2\right),\left(2,3\right)$ and $\left(3,1\right)$ is $\left(\alpha ,\beta \right)$, then the quadratic equation whose roots are $\alpha +4\beta$ and $4\alpha +\beta$, is

The combined equation of the two lines $ax+by+c=0$ and $a^{\text{'}}x+b^{\text{'}}y+c^{\text{'}}=0$ can be written as $\left(ax+by+c\right)\left(a^{\text{'}}x+b^{\text{'}}y+c^{\text{'}}\right)=0$. The equation of the angle bisectors of the lines represented by the equation $2x^2+xy-3y^2=0$ is