Conic Sections

Let the foci of the ellipse $\dfrac{x^2}{16}+\dfrac{y^2}7=1$ and the hyperbola $\dfrac{x^2}{144}-\dfrac{y^2}\alpha=\dfrac1{25}$ coincide. Then the length of the latus rectum of the hyperbola is :

Let the tangent drawn to the parabola $y^2=24x$ at the point $(\alpha,\beta)$ is perpendicular to the line $2x+2y=5$. Then the normal to the hyperbola $\dfrac{x^2}{\alpha^2}-\dfrac{y^2}{\beta^2}=1$ at the point $(\alpha+4,\beta+4)$ does NOT pass through the point :

Let the function $f(x)=2x^2-\log_ex$, $x>0$, be decreasing in $(0,a)$ and increasing in $(a,4)$. A tangent to the parabola $y^2=4ax$ at a point P on it passes through the point $(8a,8a-1)$ but does not pass through the point $\left(-\dfrac1a,0\right)$. If the equation of the normal at P is $\dfrac x\alpha+\dfrac y\beta=1$, then $\alpha+\beta$ is equal to ______.

Let P and Q be any points on the curves $(x-1)^2+(y+1)^2=1$ and $y=x^2$, respectively. The distance between P and Q is minimum for some value of the abscissa of P in the interval

The acute angle between the pair of tangents drawn to the ellipse $2x^2+3y^2=5$ from the point $(1,3)$ is

The equation of a common tangent to the parabolas $y=x^2$ and $y=-(x-2)^2$ is

If the line $x-1=0$ is a directrix of the hyperbola $kx^2-y^2=6$, then the hyperbola passes through the point

The area of the smaller region enclosed by the curves $y^2=8x+4$ and $x^2+y^2+4\sqrt3x-4=0$ is equal to

Let $P(a,b)$ be a point on the parabola $y^2=8x$ such that the tangent at P passes through the centre of the circle $x^2+y^2-10x-14y+65=0$. Let A be the product of all possible values of $a$ and B be the product of all possible values of $b$. Then the value of $A+B$ is equal to

An ellipse $E:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ passes through the vertices of the hyperbola $H:\dfrac{x^2}{49}-\dfrac{y^2}{64}=-1$. Let the major and minor axes of the ellipse E coincide with the transverse and conjugate axes of the hyperbola H, respectively. Let the product of the eccentricities of E and H be $\dfrac12$. If $l$ is the length of the latus rectum of the ellipse E, then the value of $113l$ is equal to ______.

If the length of the latus rectum of the ellipse $x^2+4y^2+2x+8y-\lambda=0$ is 4, and $l$ is the length of its major axis, then $\lambda+l$ is equal to ______.

If the length of the latus rectum of a parabola, whose focus is $(a,a)$ and the tangent at its vertex is $x+y=a$, is 16, then $|a|$ is equal to :

A common tangent T to the curves $C_1:\dfrac{x^2}4+\dfrac{y^2}9=1$ and $C_2:\dfrac{x^2}{42}-\dfrac{y^2}{143}=1$ does not pass through the fourth quadrant. If T touches $C_1$ at $(x_1,y_1)$ and $C_2$ at $(x_2,y_2)$, then $|2x_1+x_2|$ is equal to ______.

If the tangents drawn at the points P and Q on the parabola $y^2=2x-3$ intersect at the point R(0, 1), then the orthocentre of the triangle PQR is :

For the hyperbola $H:x^2-y^2=1$ and the ellipse $E:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$, $a>b>0$, let the

(1) eccentricity of E be reciprocal of the eccentricity of H, and

(2) the line $y=\sqrt{\dfrac52}x+K$ be a common tangent of E and H.

Then $4\left(a^2+b^2\right)$ is equal to ______.

Let the hyperbola $H:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ pass through the point $\left(2\sqrt2,-2\sqrt2\right)$. A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is e times the length of the latus rectum of H, where e is the eccentricity of H, then which of the following points lies on the parabola ?

Let the tangents at the points P and Q on the ellipse $\dfrac{x^2}2+\dfrac{y^2}4=1$ meet at the point $R\left(\sqrt2,2\sqrt2-2\right)$. If S is the focus of the ellipse on its negative major axis, then $SP^2+SQ^2$ is equal to ______.

Two tangent lines $l_1$ and $l_2$ are drawn from the point (2, 0) to the parabola $2y^2=-x$. If the lines $l_1$ and $l_2$ are also tangent to the circle $(x-5)^2+y^2=r$, then 17r is equal to ______.

Let a line L pass through the point of intersection of the lines $bx+10y-8=0$ and $2x-3y=0$, $b\in\mathbf R-\left\{\frac43\right\}$. If the line L also passes through the point (1, 1) and touches the circle $17(x^2+y^2)=16$, then the eccentricity of the ellipse $\dfrac{x^2}5+\dfrac{y^2}{b^2}=1$ is :

Let the focal chord of the parabola P : $y^2=4x$ along the line L : $y=mx+c$, m > 0 meet the parabola at the points M and N. Let the line L be a tangent to the hyperbola H : $x^2-y^2=4$. If O is the vertex of P and F is the focus of H on the positive $x$-axis, then the area of the quadrilateral OMFN is :