Straight Lines — JEE Advanced PYQs

Let $S$ denote the locus of the point of intersection of the pair of lines

where $\alpha$ varies over the set of non-zero real numbers. Let $T$ be the tangent to $S$ passing through the …

Consider a triangle Δ whose two sides lie on the x-axis and the line x+y+1=0. If the orthocenter of Δ is (1, 1), then the equation of the circle passing through the vertices of the triangle Δ is

Consider the lines $L_1$ and $L_2$ defined by
$L_1:x\sqrt{2}+y-1=0$ and $L_2:x\sqrt{2}-y+1=0$
For a fixed constant $\lambda$, let C be the locus of a point P such that the product of the distance of P from $L_1$ and the distance of P from $L_2$ is ${\lambda }^2$. The …

Consider the lines $L_1$ and $L_2$ defined by
$L_1:x\sqrt{2}+y-1=0$ and $L_2:x\sqrt{2}-y+1=0$
For a fixed constant $\lambda$, let C be the locus of a point P such that the product of the distance of P from $L_1$ and the distance of P from $L_2$ is ${\lambda }^2$. The …

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 …

For $a>b>c>0$, the distance between $(1,1)$ and the point of intersection of the lines $ax+by+c=0$ and $bx+ay+c=0$ is less than $2\sqrt{2}$. Then

A straight line $L$ through the point $(3,-2)$ is inclined at an angle $60^{\circ }$ to the line $\sqrt{3}x+y=1$. If $L$ also intersects the $x$-axis, then the equation of $L$ is

The locus of the orthocentre of the triangle formed by the lines

and $y=0$, where $p\ne q$, is

Let $a$ and $b$ be non-zero real numbers. Then, the equation

represents

Consider three points $P=(-\sin (\beta -\alpha ),-\cos \beta )$, $Q=(\cos (\beta -\alpha ),\sin \beta )$ and $R=(\cos (\beta -\alpha +\theta ),\sin (\beta -\theta ))$, where $0<\alpha ,\beta ,\theta <\frac{\pi }{4}$. Then,

Consider the lines given by

Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate …

Let $O(0,0),P(3,4),Q(6,0)$ be the vertices of the triangle $OPQ$. The point $R$ inside the triangle $OPQ$ is such that the triangles $OPR,PQR,OQR$ are of equal area. The coordinates of $R$ are

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 $2\sqrt{2}:\sqrt{5}$. …