Straight Lines

The line $y=x+1$ meets the ellipse $\dfrac{x^2}4+\dfrac{y^2}2=1$ at two points $P$ and $Q$. If $r$ is the radius of the circle with $PQ$ as diameter then $(3r)^2$ is equal to :

Let R be the point $(3,7)$ and let P and Q be two points on the line $x+y=5$ such that PQR is an equilateral triangle. Then the area of $\Delta PQR$ is :

Let C be a circle passing through the points A$(2,-1)$ and B$(3,4)$. The line segment AB is not a diameter of C. If r is the radius of C and its centre lies on the circle $(x-5)^2+(y-1)^2=\dfrac{13}2$, then $r^2$ is equal to :

The locus of the mid point of the line segment joining the point $(4,3)$ and the points on the ellipse $x^2+2y^2=4$ is an ellipse with eccentricity :

Let a line $L_1$ be tangent to the hyperbola $\dfrac{x^2}{16}-\dfrac{y^2}4=1$ and let $L_2$ be the line passing through the origin and perpendicular to $L_1$. If the locus of the point of intersection of $L_1$ and $L_2$ is $\left(x^2+y^2\right)^2=\alpha x^2+\beta y^2$, then $\alpha+\beta$ is equal to ______.

In an isosceles triangle ABC, the vertex A is $(6,1)$ and the equation of the base BC is $2x+y=4$. Let the point B lie on the line $x+3y=7$. If $(\alpha,\beta)$ is the centroid of $\Delta ABC$, then $15(\alpha+\beta)$ is equal to :

A rectangle R with end points of one of its sides as $(1,2)$ and $(3,6)$ is inscribed in a circle. If the equation of a diameter of the circle is $2x-y+4=0$, then the area of R is ______.

If the equation of the parabola, whose vertex is at $(5,4)$ and the directrix is $3x+y-29=0$, is $x^2+ay^2+bxy+cx+dy+k=0$, then $a+b+c+d+k$ is equal to

Let a circle C of radius 5 lie below the $x$-axis. The line $L_1:4x+3y+2=0$ passes through the centre P of the circle C and intersects the line $L_2:3x-4y-11=0$ at Q. The line $L_2$ touches C at the point Q. Then the distance of P from the line $5x-12y+51=0$ is ______.

A ray of light passing through the point $P(2,3)$ reflects on the $x$-axis at point A and the reflected ray passes through the point $Q(5,4)$. Let R be the point that divides the line segment AQ internally into the ratio 2:1. Let the co-ordinates of the foot of the perpendicular M from R on the bisector of the angle PAQ be $(\alpha,\beta)$. Then, the value of $7\alpha+3\beta$ is equal to ______.

Let $l$ be a line which is normal to the curve $y=2x^2+x+2$ at a point P on the curve. If the point $Q(6,4)$ lies on the line $l$ and O is origin, then the area of the triangle OPQ is equal to ______.

Let the lines $y+2x=\sqrt{11}+7\sqrt7$ and $2y+x=2\sqrt{11}+6\sqrt7$ be normal to a circle $C:(x-h)^2+(y-k)^2=r^2$. If the line $\sqrt{11}y-3x=\dfrac{5\sqrt{77}}3+11$ is tangent to the circle C, then the value of $(5h-8k)^2+5r^2$ is equal to ______.

Let a triangle be bounded by the lines $L_1:2x+5y=10$; $L_2:-4x+3y=12$ and the line $L_3$, which passes through the point $P(2,3)$, intersects $L_2$ at A and $L_1$ at B. If the point P divides the line-segment AB, internally in the ratio $1:3$, then the area of the triangle is equal to :

If vertex of a parabola is $(2,-1)$ and the equation of its directrix is $4x-3y=21$, then the length of its latus rectum is :

The distance between the two points A and A' which lie on $y=2$ such that both the line segments AB and A'B (where B is the point $(2,3)$) subtend angle $\dfrac\pi4$ at the origin, is equal to :

The distance of the origin from the centroid of the triangle whose two sides have the equations $x-2y+1=0$ and $2x-y-1=0$ and whose orthocenter is $\left(\dfrac73,\dfrac73\right)$ is:

If $xy^4$ attains maximum value at the point $(x,y)$ on the line passing through the points $(50+\alpha,0)$ and $(0,50+\alpha)$, $\alpha>0$, then $(x,y)$ also lies on the line :

Let $\alpha_1,\alpha_2$ $(\alpha_1<\alpha_2)$ be the values of $\alpha$ for the points $(\alpha,-3)$, $(2,0)$ and $(1,\alpha)$ to be collinear. Then the equation of the line, passing through $(\alpha_1,\alpha_2)$ and making an angle of $\dfrac\pi3$ with the positive direction of the $x$-axis, is :

Consider three circles :
$$C_1:x^2+y^2=r^2$$
$$C_2:(x-1)^2+(y-1)^2=r^2$$
$$C_3:(x-2)^2+(y-1)^2=r^2$$
If a line $L:y=mx+c$ be a common tangent to $C_1$, $C_2$ and $C_3$ such that $C_1$ and $C_3$ lie on one side of line L while $C_2$ lies on other side, then the value of $20\left(r^2+c\right)$ is equal to :

A line, with the slope greater than one, passes through the point $A(4,3)$ and intersects the line $x-y-2=0$ at the point B. If the length of the line segment AB is $\dfrac{\sqrt{29}}3$, then B also lies on the line :