Conic Sections Questions (164)

Length of latus rectum of the parabola is:

$\min \left[(x_1 - x_2)^2 + \left|12 - \sqrt{1 - x_1^2} - \sqrt{4x_2}\right|^2\right], \forall x_1, x_2 \in \mathbb{R}$ is:

Let $P(x, y)$ is a variable point such that $\sqrt{(x-1)^2 + (y-2)^2} - \sqrt{(x-5)^2 + (y-5)^2} = 3$ which represents hyperbola. The eccentricity $e'$ of the corresponding conjugate hyperbola is:

$(x-1)(y-2) = 5$ and $(x-1)^2 + (y+2)^2 = r^2$ intersect at four points $A, B, C, D$ and if centroid of $\triangle ABC$ lies on line $y = 3x - 4$, then locus of $D$ is:

For which of the following hyperbolas, we can have more than one pair of perpendicular tangents?

If $P, Q$ are ends of focal chord of the parabola, then $\frac{1}{SP} + \frac{1}{SQ} =$

Tangents are drawn to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b)$ and the circle $x^2 + y^2 = a^2$ at the points where a common ordinate cuts them (on the same side of the $x$-axis). Then the greatest acute angle between these tangents is given by:

Let $A\left(\frac{1}{2}, 0\right), B\left(\frac{3}{2}, 0\right), C\left(\frac{5}{2}, 0\right)$ be the given points and $P$ be a point satisfying $\max(PA + PB, PB + PC) < 2$. All points $P$ are points common to:

If origin is shifted to point $\left(3, \frac{7}{2}\right)$ and the axes are rotated through an angle $\theta$ in clockwise sense so that equation of given hyperbola changes to the standard form $\frac{x'^2}{a^2} - \frac{y'^2}{b^2} = 1$, then $\theta$ is:

An equilateral triangle $SAB$ is inscribed in the parabola $y^2 = 4ax$ having its focus at '$S$'. If chord $AB$ lies towards the left of $S$, then side length of this triangle is:

If a ray of light incident along the line $3x + (5 - 4\sqrt{2})y = 15$ gets reflected from the hyperbola $\frac{x^2}{16} - \frac{y^2}{9} = 1$ at the point $(4\sqrt{2}, 3)$, then its reflected ray goes along the line:

The locus of a point on the variable parabola $y^2 = 4ax$, whose distance from focus is constant $k$, is equal to: ($a$ is parameter)

If $P = (x_1, y_1)$ the slope of third normal is:

If the eccentricity of the hyperbola $x^2 - y^2\sec^2 a = 5$ is $\sqrt{3}$ times the eccentricity of the ellipse $x^2\sec^2 a + y^2 = 25$, then a value of $a$ is: