Coordinate Geometry

If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is:

Let the point $P$ of the focal chord $PQ$ of the parabola $y^2 = 16x$ be $(1, -4)$. If the focus of the parabola divides the chord $PQ$ in the ratio $m : n$, $\gcd(m, n) = 1$, then $m^2 + n^2$ is equal to:

Let the area of the triangle formed by a straight line $L : x + by + c = 0$ with coordinate axes be $48$ square units. If the perpendicular drawn from the origin to the line $L$ makes an angle of $45^\circ$ with the positive x-axis, then the value of $b^2 + c^2$ is:

Let one focus of the hyperbola $H:\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}=1$ be at $(\sqrt{10},0)$ and the corresponding directrix be $x=\tfrac{9}{\sqrt{10}}$. If $e$ and $l$ respectively are the eccentricity and the length of the latus rectum of $H$, then $9\,(e^2+l)$ is equal to:

If $S$ and $S'$ are the foci of the ellipse $\frac{x^2}{18} + \frac{y^2}{9} = 1$ and $P$ is a point on the ellipse, then $\min\bigl(SP\cdot S'P\bigr) + \max\bigl(SP\cdot S'P\bigr)$ is equal to:

Let the focal chord $PQ$ of the parabola $y^2 = 4x$ make an angle of $60^\circ$ with the positive x-axis, where $P$ lies in the first quadrant. If the circle, whose one diameter is $PS$, $S$ being the focus of the parabola, touches the y-axis at the point $(0,\alpha)$, then $5\alpha^2$ is equal to:

If the four distinct points $(4,6)$, $(-1,5)$, $(0,0)$ and $(k,3k)$ lie on a circle of radius $r$, then $10k + r^2$ is equal to:

Consider the lines $x(3\lambda+1)+y(7\lambda+2)=17\lambda+5$, $\lambda$ being a parameter, all passing through a point $P$. One of these lines (say $L$) is farthest from the origin. If the distance of $L$ from the point $(3,6)$ is $d$, then the value of $d^2$ is:

Let the equation $x(x+2)(12-k)=2$ have equal roots. Then the distance of the point $(k,\tfrac{k}{2})$ from the line $3x+4y+5=0$ is:

The shortest distance between the curves $y^2 = 8x$ and $x^2 + y^2 + 12y + 35 = 0$ is: