Conic Sections

If a tangent to a suitable conic (Column 1) is found to be $y = x + 8$ and its point of contact is $(8, 16)$, then which of the following options is the only CORRECT combination?

The tangent to a suitable conic (Column 1) at $\left(\sqrt{3}, \frac{1}{2}\right)$ is found to be $\sqrt{3}x + 2y = 4$, then which of the following options is the only CORRECT combination?

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

The eccentricity of an ellipse whose centre is at the origin is $\dfrac12$. If one of its directrices is $x=-4$, then the equation of the normal to it at $\left(1,\dfrac32\right)$ is :

A hyperbola passes through the point $P\left(\sqrt2,\sqrt3\right)$ and has foci at $(\pm2,0)$. Then the tangent to this hyperbola at P also passes through the point :

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

If the common tangents to the parabola, $x^2=4y$ and the circle, $x^2+y^2=4$ intersect at the point P, then the distance of P from the origin, is :

Consider an ellipse, whose centre is at the origin and its major axis is along the $x$-axis. If its eccentricity is $\dfrac35$ 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 having centre at the origin, axes along the co-ordinate axes and passing through the points (4, −1) and (−2, 2) is :

If $y=mx+c$ is the normal at a point on the parabola $y^2=8x$ whose focal distance is 8 units, then |c| is equal to :

A solution curve of the differential equation $(x^2+xy+4x+2y+4)\dfrac{dy}{dx}-y^2=0$, $x>0$, passes through the point $(1,3)$. Then the solution curve

The circle $C_1:x^2+y^2=3$, with centre at $O$, intersects the parabola $x^2=2y$ at the point $P$ in the first quadrant. Let the tangent to the circle $C_1$ at $P$ touches other two circles $C_2$ and $C_3$ at $R_2$ and $R_3$, respectively. Suppose $C_2$ and $C_3$ have equal radii $2\sqrt3$ and centres $Q_2$ and $Q_3$, respectively. If $Q_2$ and $Q_3$ lie on the $y$-axis, then

Let $P$ be the point on the parabola $y^2=4x$ which is at the shortest distance from the center $S$ of the circle $x^2+y^2-4x-16y+64=0$. Let $Q$ be the point on the circle dividing the line segment $SP$ internally. Then

The orthocentre of the triangle $F_1MN$ is

If the tangents to the ellipse at $M$ and $N$ meet at $R$ and the normal to the parabola at $M$ meets the $x$-axis at $Q$, then the ratio of area of the triangle $MQR$ to area of the quadrilateral $MF_1NF_2$ is

The centres of those circles which touch the circle, $x^2+y^2-8x-8y-4=0$, externally and also touch the $x$-axis, lie on :

Let P be the point on the parabola, $y^2=8x$ which is at a minimum distance from the centre C of the circle, $x^2+(y+6)^2=1$. Then the equation of the circle, passing through C and having its centre at P is :

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is :

Let a and b respectively be the semi-transverse and semi-conjugate axes of a hyperbola whose eccentricity satisfies the equation $9e^2-18e+5=0$. 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 :

If the tangent at a point on the ellipse $\dfrac{x^2}{27}+\dfrac{y^2}3=1$ meets the coordinate axes at A and B, and O is the origin, then the minimum area (in sq. units) of the triangle OAB is :