Mock Test 2 — Hyperbola

Let $P(a\sec\theta, b\tan\theta)$ be a variable point on the hyperbola. The normal line at $P$ internally bisects the focal angle $\angle SPS'$. The reflection property dictates that an incident ray directed towards $S$ will reflect along a path that:

Find the equation of the common tangent line to the parabola $y^2 = 8x$ and the standard hyperbola $\ds\frac{x^2}{1} - \frac{y^2}{4} = 1$:

If a variable chord of the rectangular hyperbola $xy = c^2$ is bisected exactly at a moving coordinate point $M(h, k)$ such that it maintains a constant slope tracking of $m = -2$, the locus of $M$ forms a:

Find the coordinate position of the point of intersection of the two tangents drawn at the parameter points $\theta_1$ and $\theta_2$ on the standard hyperbola:

If the normal line drawn at the point $P(6, 3)$ on the hyperbola $\ds\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ intersects the horizontal principal x-axis at $(9, 0)$, calculate the exact eccentricity $e$ of the conic profile:

The product of the lengths of the perpendicular distances dropped from any point on the hyperbola $\ds\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ to its two asymptotes is constant and identically equal to:

If the tangent line drawn at parameter point $t$ on the rectangular hyperbola $xy = c^2$ cuts off intercept segments $OA$ and $OB$ on the coordinate axes, the midpoint of the intercepted segment $AB$ matches which position?

Find the joint equation of the pair of asymptotes for the shifted hyperbola profile given by $xy - 2x - 3y = 0$:

The director circle of the rectangular hyperbola $x^2 - y^2 = a^2$ collapses structurally into:

If the line $lx + my + n = 0$ is a valid normal line to the standard hyperbola $\ds\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, it must satisfy which coefficient condition?

If the four normals drawn at the parameter points $t_1, t_2, t_3,$ and $t_4$ on the rectangular hyperbola $xy = c^2$ are concurrent, find the value of the parameter product $t_1 t_2 t_3 t_4$.

Find the total number of real common tangents that can be constructed to the two non-intersecting conics $x^2 + y^2 = 1$ and $\ds\frac{x^2}{4} - y^2 = 1$.

If the angle between the two asymptotes of a hyperbola is $60^\circ$, and its eccentricity is $e$, calculate the value of $3e^2$ if the hyperbola is a conjugate profiling type ($b > a$).

Assertion (A): The segment of any variable tangent line to a hyperbola intercepted between its two asymptotes is always bisected exactly at its point of contact.
Reason (R): Formulating the parametric line metrics confirms that the coordinate shifts match the harmonic means across the boundary thresholds.

Assertion (A): The variation of the parametric string property for a hyperbola dictates that $|S'P - SP| = 2a$ for all points $P$ on its boundary curves.
Reason (R): The focus-directrix mapping tracks the distance ratio as $e > 1$, which algebraically transforms into linear expressions in terms of the abscissa of $P$.