Straight Line and a Point

The number of lines that can be drawn from the point $\left(2,3\right)$, so that its distance from $\left(-1,6\right)$ is equal to $6$ is

If the image of point $\left(4,-3\right)$ with respect to the line $y=x$ is $\left(a,b\right)$, then find $a+b$.

Let $S$ be a subset of the plane defined by $S=\left\{\left(x,y\right):\left|x\right|+2\left|y\right|=1\right\}$. Then, the radius of the smallest circle with centre at the origin and having non-empty intersection with $S$ is

The point of intersection of the pair of lines $x^2+xy+2y^2-3x+2y+4=0$ is

If one of the lines $2x^2-xy+b{y}^2=0$ passes through the point $\left(-4,-2\right),$ then $b^2=$

Consider the plane $x+y-z=1$ and point $A(1,2,-3)$ . A line $L$ has the equation $x=1+3r, y=2-r \& z=3+4r.$

 The distance between the points on the line which are at a distance of $4/\sqrt{3}$  from the plane is

Consider the triangle having vertices $O\left(0, 0\right), A\left(2, 0\right)$ and $B\left(1, \sqrt{3}\right)$. Also $b\le min\left\{a_1, a_2, a_3\right\}$ means $b\le a_1$ when $a_1$ is least ; $b\le a_2$ when $a_2$ is least and so on. From this we can say $b\le a_1, b\le a_2, ........., b\le a_n$ .

For next two question please follow the same

Consider the triangle having vertices $O\left(0, 0\right), A\left(2, 0\right)$ and $B\left(1, \sqrt{3}\right)$. Also $b\le min\left\{a_1, a_2, a_3\right\}$ means $b\le a_1$ when $a_1$ is least ; $b\le a_2$ when $a_2$ is least and so on. From this we can say $b\le a_1, b\le a_2, ........., b\le a_n$ .

Let $R$ be the region consisting of all those points $P$ inside $\Delta \text{OAB}$ which satisfy $\text{d}\left(\text{P, OA}\right)\le min[\text{d}\left(\text{P, OB}\right)\le \text{d}\left(\text{P, AB}\right)]$, where $d$ denotes the distance from the point to the corresponding line. Then the area of the regiona$R$ is

The equation of the image of line $y=x$ with respect to the line mirror $2x-y=1$ is

The lengths of the perpendiculars from the points $\left(m^2,2m\right), \left(mn,m+n\right)$ and $\left(n^2,2n\right)$ to the line $x+\sqrt{3}y+3=0$ are in

Complete the following table :

Complete the following table :

Image of point $\left(5,2,-7\right)$ in $xy-$plane is given by $\left(x,y,z\right)$. Find product $xyz$.

Let $A(0, 1)$ and $B(2, 0)$, and $P$ be a point on the line $4x+3y=-9$.Then the coordinates of $P$ such that $\left|AP-PB\right|$ is maximum is:

The lines $2x+y=5$ and $x+2y=4$ intersect at the point:

The straight lines $x+y=0, 5x+y=4$ and $x+5y=4$ forms a triangle which is?

The maximum possible length of semi latus rectum is

If the point $(a, a)$ falls between the lines $|x+y|=2,$ then :

The combined equation of three sides of a triangle is $\left(x^2-y^2\right)(2x+3y-6)=0.$ If $(-2, a)$ is an interior point and $\left(b, 1\right)$ is an exterior point of the triangle, then