JEE Main 2017 — Coordinate Geometry

Let $k$ be an integer such that the triangle with vertices $(k,-3k), (5,k)$ and $(-k, 2)$ has area $28$ sq. units. Then the orthocenter of this triangle is at the point:

A square, of each side $2$, lies above the $x$-axis and has one vertex at the origin. If one of the sides passing through the origin makes an angle $30^{\circ}$ with the positive direction of the $x$-axis , then the sum of the $x$-coordinates of the vertices of the square is$:$

Consider an ellipse, whose center is at the origin and its major axis is along the $x$-axis. If its eccentricity is $\frac{3}{5}$ and the distance between its foci is $6$, then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is:

The eccentricity of an ellipse whose centre is at the origin is $\frac{1}{2}$ . If one of its directrices is $x=-4$ , then the equation of the normal to it at $(1,\frac{3}{2})$ is:

A line drawn through the point $P(4,7)$ cuts the circle ${x}^{2}+{y}^{2}=9$ at the points $A$ and $B$. Then ${P}_{A}\cdot {P}_{B}$ is equal to.

The eccentricity of an ellipse having centre at the origin, axes along the co-ordinate axes and passing through the points $(4,-1)$ and $(-2,2)$ is

The radius of a circle, having minimum area, which touches the curve $y=4-{x}^{2}$ and the lines, $y=|x|$ is:

If two parallel chords of a circle, having diameter $4$ units, lie on the opposite sides of the center and subtend angles ${\mathrm{cos}}^{-1}⁡(\frac{1}{7})$ and ${\mathrm{sec}}^{-1}⁡(7)$ at the center respectively, then the distance between these chords is:

If a point $P(0,-2)$ and $Q$ is any point on the circle, ${x}^{2}+{y}^{2}-5x-y+5=0$, then the maximum value of ${(PQ)}^{2}$ is

The locus of the point of intersection of the straight lines, $tx-2y-3t=0$ and $x-2ty+3=0 (t\in R),$ is: