Straight Lines

Locus of the image of the point (2, 3) in the line $(2x-3y+4)+k(x-2y+3)=0$, $k\in\mathbb R$, is a :

Let O be the vertex and Q be any point on the parabola, $x^2=8y$. If the point P divides the line segment OQ internally in the ratio 1 : 3, then the locus of P is :

The largest value of r for which the region represented by the set $\{\omega\in\mathbb C/|\omega-4-i|\le r\}$ is contained in the region represented by the set $\{z\in\mathbb C/|z-1|\le|z+i|\}$, is equal to :

The points $\left(0,\dfrac83\right)$, (1, 3) and (82, 30) :

Let L be the line passing through the point P(1, 2) such that its intercepted segment between the co-ordinate axes is bisected at P. If $L_1$ is the line perpendicular to L and passing through the point (−2, 1), then the point of intersection of L and $L_1$ is :

A straight line L through the point (3, −2) is inclined at an angle 60° to the line $\sqrt3x+y=1$. If L also intersects the $x$-axis, then the equation of L is :

If the incentre of an equilateral triangle is (1, 1) and the equation of its one side is $3x+4y+3=0$, then the equation of the circumcircle of this triangle is :

Let PQ be a double ordinate of the parabola, $y^2=-4x$, where P lies in the second quadrant. If R divides PQ in the ratio 2 : 1, then the locus of R is :

For a point $P$ in the plane, let $d_1(P)$ and $d_2(P)$ be the distances of the point $P$ from the lines $x-y=0$ and $x+y=0$ respectively. The area of the region $R$ consisting of all points $P$ lying in the first quadrant of the plane and satisfying $2\le d_1(P)+d_2(P)\le4$, is

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 :

Let $a$, $b$, $c$ and $d$ be non-zero numbers. If the point of intersection of the lines $4ax+2ay+c=0$ and $5bx+2by+d=0$ lies in the fourth quadrant and is equidistant from the two axes then :

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

Let $a$ and $b$ be any two numbers satisfying $\dfrac1{a^2}+\dfrac1{b^2}=\dfrac14$. Then, the foot of perpendicular from the origin on the variable line, $\dfrac xa+\dfrac yb=1$, lies on :

Given three points P, Q, R with P(5, 3) and R lies on the $x$-axis. If 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 stair-case of length $l$ rests against a vertical wall and a floor of a room,. Let 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 stair-case begins to slide on the floor, then the locus of P is :

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

Let $A(2,3,5)$, $B(-1,3,2)$ and $C(\lambda,5,\mu)$ be the vertices of a $\Delta ABC$. If the median through A is equally inclined to the coordinate axes, then :

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

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

Two tangents are drawn from a point $(-2,-1)$ to the curve, $y^2=4x$. If $\alpha$ is the angle between them, then $|\tan\alpha|$ is equal to :