Straight Lines

Consider
$$L_1:2x+3y+p-3=0,\qquad L_2:2x+3y+p+3=0,$$
where $p$ is a real number, and $C:x^2+y^2+6x-10y+30=0$.

STATEMENT-1: If line $L_1$ is a chord of circle $C$, then line $L_2$ is not always a diameter of circle $C$.

and

STATEMENT-2: If line $L_1$ is a diameter of circle $C$, then line $L_2$ is not a chord of circle $C$.

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.

The tangent to the curve $y=e^x$ drawn at the point $(c,e^c)$ intersects the line joining the points $(c-1,e^{c-1})$ and $(c+1,e^{c+1})$

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

Let $ABCD$ be a quadrilateral with area 18, with side $AB$ parallel to the side $CD$ and $AB=2CD$. Let $AD$ be perpendicular to $AB$ and $CD$. If a circle is drawn inside the quadrilateral $ABCD$ touching all the sides, then its radius is

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.