JEE Main 2014 — Coordinate Geometry

If $OB$ is the semi-minor axis of an ellipse, ${F}_{1}$ and ${F}_{2}$ are its focii and the angle between ${F}_{1}B$ and ${F}_{2}B$ is a right angle, then the square of the eccentricity of the ellipse is

Given three points $P,Q,R$ with $P(5,3)$ and $R$ lies on the $x-$axis. If the equation of $RQ$ is $x-2y=2$ and $PQ$ is parallel to the $x-$axis, then the centroid of $\Delta PQR$ lies on the line

A chord is drawn through the focus of the parabola ${ y⁡ }^{2} = 6 x⁡$ such that its distance from the vertex of this parabola is $\frac{ \sqrt{ 5 } }{ 2 }$, then its slope can be

A stair-case of length $l$ rests against a vertical wall and a floor of a room. Let $\mathrm{P}$ be a point on the stair-case, nearer to its end on the wall, that divides its length in the ratio $1: 2$. If the staircase begins to slide on the floor, then the locus of $\mathrm{P}$ is:

If a line intercepted between the coordinate axes is trisected at a point $\mathrm{A}(4,3)$, which is nearer to $x$-axis, then its equation is:

The circumcentre of a triangle lies at the origin and its centroid is the midpoint of the line segment joining the points $({a}^{2}+1,{a}^{2}+1)$ and $(2a, - 2a),a\neq 0$. Then for any $a,$ the orthocentre of this triangle lies on the line

For the two circles $x^2+y^2=16$ and $x^2+y^2-2 y=0$, there is/are

The set of all real values of $\lambda$ for which exactly two common tangents can be drawn to the circles $x^2+y^2-4 x-4 y+6=0$ and $\mathrm{x}^2+\mathrm{y}^2-10 \mathrm{x}-10 \mathrm{y}+\lambda=0$ is the interval:

Let $\mathrm{L}_1$ be the length of the common chord of the curves $x^2+y^2=9$ and $y^2=8 x$, and $L_2$ be the length of the latus rectum of $y^2=8 x$, then:

If the point $(1,4)$ lies inside the circle ${x}^{2}+{y}^{2}-6x+10y+p=0$ and the circle does not touch or intersect the coordinate axes, then the set of all possible values of $p$ is the interval

Let $a$ and $b$ be any two numbers satisfying $\frac{1}{{a}^{2}}+\frac{1}{{b}^{2}}=\frac{1}{4}.$ Then, the foot of perpendicular from the origin on the variable line $\frac{x}{a}+\frac{y}{b}=1$ lies on :

The locus of the foot of perpendicular drawn from the centre of the ellipse ${x}^{2}+3{y}^{2}=6$ on any tangent to it is

If the three distinct lines $x+2 a y+a=0, x+3 b y$ $+b=0$ and $x+4 a y+a=0$ are concurrent, then the point $(a, b)$ lies on $a$ :

Let $PS$ be the median of the triangle with vertices $P(2,2),Q(6,-1)$ and $R(7,3)$. The equation of the line passing through $(1,-1)$and parallel to $PS$ is

The base of an equilateral triangle is along the line given by $3 x+4 y=9$. If a vertex of the triangle is $(1,2)$, then the length of a side of the triangle is:

If a line $L$ is perpendicular to the line $5x-y=1,$ and the area of the triangle formed by the line $L$ and the coordinate axes is $5$ sq units, then the distance of the line $L$ from the line $x+5y=0$ is