Straight Lines

A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is $\dfrac14$. Three stones A, B and C are placed at the points (1, 1), (2, 2) and (4, 4) respectively. Then which of these stones is/are on the path of the man ?

The locus of the mid-point of the line segment joining the focus of the parabola $y^2=4ax$ to a moving point of the parabola, is another parabola whose directrix is :

If the tangent to the curve $y=x^3$ at the point P(t, t$^3$) meets the curve again at Q, then the ordinate of the point which divides PQ internally in the ratio 1 : 2 is :

A line $y = mx + 1$ intersects the circle $(x-3)^2 + (y+2)^2 = 25$ at the points $P$ and $Q$. If the midpoint of the line segment $PQ$ has $x$-coordinate $-\dfrac35$, then which one of the following options is correct?

Let the point $B$ be the reflection of the point $A(2, 3)$ with respect to the line $8x - 6y - 23 = 0$. Let $\Gamma_A$ and $\Gamma_B$ be circles of radii 2 and 1 with centres $A$ and $B$ respectively. Let $T$ be a common tangent to the circles $\Gamma_A$ and $\Gamma_B$ such that both the circles are on the same side of $T$. If $C$ is the point of intersection of $T$ and the line passing through $A$ and $B$, then the length of the line segment $AC$ is ______

Let $P$ be a point on the circle $S$ with both coordinates being positive. Let the tangent to $S$ at $P$ intersect the coordinate axes at the points $M$ and $N$. Then, the mid-point of the line segment $MN$ must lie on the curve

Let $T$ be the line passing through the points $P(-2, 7)$ and $Q(2, -5)$. Let $F_1$ be the set of all pairs of circles $(S_1, S_2)$ such that $T$ is tangent to $S_1$ at $P$ and tangent to $S_2$ at $Q$, and also such that $S_1$ and $S_2$ touch each other at a point, say, $M$. Let $E_1$ be the set representing the locus of $M$ as the pair $(S_1, S_2)$ varies in $F_1$. Let the set of all straight line segments joining a pair of distinct points of $E_1$ and passing through the point $R(1, 1)$ be $F_2$. Let $E_2$ be the set of the mid-points of the line segments in the set $F_2$. Then, which of the following statement(s) is (are) TRUE?

Let k be an integer such that the triangle with vertices $(k,-3k)$, $(5,k)$ and $(-k,2)$ has area 28 sq. units. Then the orthocentre of this triangle is at the point :

The locus of the point of intersection of the straight lines,
$$tx-2y-3t=0$$
$$x-2ty+3=0\quad(t\in\mathbb R),$$
is :

A square, of each side 2, lies above the $x$-axis and has one vertex at the origin. If one of the sides passing through the origin makes an angle 30° with the positive direction of the $x$-axis, then the sum of the $x$-coordinates of the vertices of the square is :

Let $RS$ be the diameter of the circle $x^2+y^2=1$, where $S$ is the point $(1,0)$. Let $P$ be a variable point (other than $R$ and $S$) on the circle and tangents to the circle at $S$ and $P$ meet at the point $Q$. The normal to the circle at $P$ intersects a line drawn through $Q$ parallel to $RS$ at point $E$. Then the locus of $E$ passes through the point(s)

Two sides of a rhombus are along the lines, $x-y+1=0$ and $7x-y-5=0$. If its diagonals intersect at (−1, −2), then which one of the following is a vertex of this rhombus ?

If a variable line drawn through the intersection of the lines $\dfrac x3+\dfrac y4=1$ and $\dfrac x4+\dfrac y3=1$, meets the coordinate axes at A and B, (A ≠ B), then the locus of the midpoint of AB is :

The point (2, 1) is translated parallel to the line L : x − y = 4 by $2\sqrt3$ units. If the new point Q lies in the third quadrant, then the equation of the line passing through Q and perpendicular to L is :

In a triangle ABC, right angled at the vertex A, if the position vectors of A, B and C are respectively $3\hat i+\hat j-\hat k$, $-\hat i+3\hat j+p\hat k$ and $5\hat i+q\hat j-4\hat k$, then the point (p, q) lies on a line :

A ray of light is incident along a line which meets another line, 7x − y + 1 = 0, at the point (0, 1). The ray is then reflected from this point along the line, y + 2x = 1. Then the equation of the line of incidence of the ray of light is :

A straight line through origin O meets the lines 3y = 10 − 4x and 8x + 6y + 5 = 0 at points A and B respectively. Then O divides the segment AB in the ratio :

Let the curve $C$ be the mirror image of the parabola $y^2=4x$ with respect to the line $x+y+4=0$. If $A$ and $B$ are the points of intersection of $C$ with the line $y=-5$, then the distance between $A$ and $B$ is

The normal to the curve, $x^2+2xy-3y^2=0$, at (1, 1) :

The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices (0, 0), (0, 41) and (41, 0), is :