JEE Main 2016 — Coordinate Geometry

A straight line through origin $O$ meets the lines $3y=10-4x$and $8x+6y+5=0$ at points $A$ and $B$ respectively. Then, $O$ divides the segment $AB$ in the ratio

Equation of the tangent to the circle, at the point $(1,-1)$, whose center, is the point of intersection of the straight lines $x-y=1$ and $2x+y=3$ is:

If a variable line drawn through the intersection of the lines $\frac{x}{3}+\frac{y}{4}=1$ and $\frac{x}{4}+\frac{y}{3}=1$ , meets the coordinate axes at $A$ and $B$, $(A \neq B),$then the locus of the midpoint of $\mathrm{AB}$ is:

If one of the diameters of the circle, given by the equation, ${x}^{2}+{y}^{2}-4x+6y-12=0,$ is a chord of a circle $S$, whose centre is at $(-3,2)$, then the radius of $S$ is

Let $a$ and $b$ respectively be the semi-transverse and semi-conjugate axes of a standard hyperbola whose eccentricity satisfies the equation $9{e}^{2}-18e+5=0$. If $S(5, 0)$ is a focus and $5x=9$ is the corresponding directrix of this hyperbola, then ${a}^{2}-{b}^{2}$ is equal to

The eccentricity of the hyperbola whose length of its conjugate axis is equal to half of the distance between its foci, is

The point $(2,1)$ is translated parallel to the line $L:x-y=4$ by $2\sqrt{3}$ units. If the new point $Q$ lies in the third quadrant, then the equation of the line passing through $Q$ and perpendicular to $L$ is

Two sides of a rhombus are along the lines, $x-y+1=0$ and $7x-y-5=0$ . If its diagonals intersect at $(-1, -2)$ , then which one of the following is a vertex of this rhombus ?