Straight Lines

Let $P$ be the point in the first quadrant where the given ellipses intersect. If $\theta$ is the acute angle between the tangents to the given ellipses at the point $P$, then the value of $4\tan\theta$ is ________.

Let the mid points of the sides of a triangle $ABC$ be $\left(\frac52,7\right)$, $\left(\frac52,3\right)$ and $(4,5)$. If its incentre is $(h,k)$, then $3h+k$ is equal to :

Let a circle $C$ have its centre in the first quadrant, intersect the coordinate axes at exactly three points and cut off equal intercepts from the coordinate axes. If the length of the chord of $C$ on the line $x+y=1$ is $\sqrt{14}$, then the square of the radius of $C$ is ________.

Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines $x+(k-1)y+3=0$ and $2x+k^2y-4=0$. If the line $x-y+2=0$ intersects the circle at the points A and B, then $(AB)^2$ is equal to :

Let O be the origin, and P and Q be two points on the rectangular hyperbola $xy=12$ such that the mid point of the line segment PQ is $\left(\dfrac12,-\dfrac12\right)$. Then the area of the triangle OPQ equals :

Let A be the point $(3,0)$ and circles with variable diameter AB touch the circle $x^2+y^2=36$ internally. Let the curve C be the locus of the point B. If the eccentricity of C is $e$, then $72e^2$ is equal to ________.

Let the line $L_1:x+3=0$ intersect the lines $L_2:x-y=0$ and $L_3:3x+y=0$ at the points A and B, respectively. Let the bisector of the obtuse angle between the lines $L_2$ and $L_3$ intersect the line $L_1$ at the point C. Then $BC^2:AC^2$ is equal to:

Let the vertex A of a triangle ABC be $(1,2)$, and the mid-point of the side AB be $(5,-1)$. If the centroid of this triangle is $(3,4)$ and its circumcenter is $(\alpha,\beta)$, then $21(\alpha+\beta)$ is equal to:

Let $P(3\cos\alpha,2\sin\alpha)$, $\alpha\ne0$, be a point on the ellipse $\dfrac{x^2}9+\dfrac{y^2}4=1$, Q be a point on the circle $x^2+y^2-14x-14y+82=0$ and R be a point on the line $x+y=5$ such that the centroid of the triangle PQR is $\left(2+\cos\alpha,\ 3+\dfrac23\sin\alpha\right)$. Then the sum of the ordinates of all possible points R is:

Let A, B be points on the two half-lines $x-\sqrt3|y|=\alpha$, $\alpha>0$ at a distance of $\alpha$ from their point of intersection P. The line segment AB meets the angle bisector of the given half-lines at the point Q. If $PQ=\dfrac92$ and R is the radius of the circumcircle of $\Delta PAB$, then $\dfrac{\alpha^2}R$ is equal to ________.

Let A, B and C be the vertices of a variable right angled triangle inscribed in the parabola $y^2=16x$. Let the vertex B containing the right angle be $(4,8)$ and the locus of the centroid of $\Delta ABC$ be a conic $C_0$. Then three times the length of latus rectum of $C_0$ is ________.

In an equilateral triangle PQR, let the vertex P be at $(3,5)$ and the side QR be along the line $x+y=4$. If the orthocentre of the triangle PQR is $(\alpha,\beta)$, then $9(\alpha+\beta)$ is equal to:

From the point $(-1,-1)$, two rays are sent making angles of 45° with the line $x+y=0$. These rays get reflected from the mirror $x+2y=1$. If the equations of the reflected rays are $ax+by=9$ and $cx+dy=7$, $a,b,c,d\in\mathbf Z$, then the value of $ad+bc$ is ________.

Let chord PQ of length $3\sqrt{13}$ of the parabola $y^2=12x$ be such that the ordinates of points P and Q are in the ratio 1:2. If the chord PQ subtends an angle $\alpha$ at the focus of the parabola, then $\sin\alpha$ is equal to:

Let $x=9$ be a directrix of an ellipse E, whose centre is at the origin and eccentricity is $\dfrac13$. Let $P(\alpha,0)$, $\alpha>0$, be a focus of E and AB be a chord passing through P. Then the locus of the mid point of AB is :

Let the line $x-y=4$ intersect the circle $C:(x-4)^2+(y+3)^2=9$ at the points Q and R. If $P(\alpha,\beta)$ is a point on C such that $PQ=PR$, then $(6\alpha+8\beta)^2$ is equal to ________.

If a straight line drawn through the point of intersection of the lines $4x+3y-1=0$ and $3x+4y-1=0$, meets the co-ordinate axes at the points P and Q, then the locus of the mid point of PQ is :

Let O be the vertex of the parabola $y^2=4x$ and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C, then the length of its latus rectum is :

Consider the circle $C:x^2+y^2-6x-8y-11=0$. Let a variable chord AB of the circle C subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from the origin on the chord AB is the circle $x^2+y^2-\alpha x-\beta y-\gamma=0$, then $\alpha+\beta+2\gamma$ is equal to ________.

For any two points $M$ and $N$ in the $XY$-plane, let $\overrightarrow{MN}$ denote the vector from $M$ to $N$, and $\vec 0$ denote the zero vector. Let $P$, $Q$ and $R$ be three distinct points in the $XY$-plane. Let $S$ be a point inside the triangle $\Delta PQR$ such that

$$\overrightarrow{SP} + 5\,\overrightarrow{SQ} + 6\,\overrightarrow{SR} = \vec 0.$$

Let $E$ and $F$ be the mid-points of the sides $PR$ and $QR$, respectively. Then the value of

$$\frac{\text{length of the line segment } EF}{\text{length of the line segment } ES}$$

is _____.