Angle between 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 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 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 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 ________.

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:

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 equations of the sides AB, BC and CA of a triangle ABC are $2x+y=0$, $x+py=15a$ and $x-y=3$ respectively. If its orthocentre is $(2,a)$, $-\dfrac12<a<2$, then p is equal to ______.

The acute angle between the pair of tangents drawn to the ellipse $2x^2+3y^2=5$ from the point $(1,3)$ is

If the tangents drawn at the points P and Q on the parabola $y^2=2x-3$ intersect at the point R(0, 1), then the orthocentre of the triangle PQR is :

Let $m_1$, $m_2$ be the slopes of two adjacent sides of a square of side a such that $a^2+11a+3(m_1^2+m_2^2)=220$. If one vertex of the square is $(10(\cos\alpha-\sin\alpha),\ 10(\sin\alpha+\cos\alpha))$, where $\alpha\in\left(0,\dfrac\pi2\right)$ and the equation of one diagonal is $(\cos\alpha-\sin\alpha)x+(\sin\alpha+\cos\alpha)y=10$, then $72(\sin^4\alpha+\cos^4\alpha)+a^2-3a+13$ is equal to :

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 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 :

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 :

If chord $PQ$ subtends an angle $\theta$ at the vertex of $y^2=4ax$, then $\tan\theta=$

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

Consider the lines given by
$$L_1:x+3y-5=0,\qquad L_2:3x-ky-1=0,\qquad L_3:5x+2y-12=0.$$
Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.

Lines $L_1:y-x=0$ and $L_2:2x+y=0$ intersect the line $L_3:y+2=0$ at $P$ and $Q$, respectively. The bisector of the acute angle between $L_1$ and $L_2$ intersects $L_3$ at $R$.

STATEMENT-1: The ratio $PR:RQ$ equals $2\sqrt2:\sqrt5$.

because

STATEMENT-2: In any triangle, bisector of an angle divides the triangle into two similar triangles.