Mock Test 1 — Hyperbola

If the line $y = mx + \sqrt{a^2m^2-b^2}$ touches the hyperbola $\ds\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ at an eccentricity boundary where $b^2 = 3a^2$, find the real range of slopes $m$ for which a real tangent line can be constructed:

Find the equation of the chord of contact drawn from the external point $P(1, 4)$ to the standard conjugate hyperbola mapping $\ds\frac{x^2}{9} - \frac{y^2}{16} = -1$:

If the eccentricities of a hyperbola and its corresponding conjugate hyperbola are $e_1$ and $e_2$ respectively, and $e_1 = \sqrt{3}$, find the value of $e_2$:

A tangent line to the hyperbola $\ds\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ at the parameter point $\theta$ cuts the asymptotes at $A$ and $B$. The geometric center $C(0,0)$ forms a triangle $CAB$ whose total area is:

Find the coordinate position of the focus of the rectangular hyperbola configuration modeled by the product formula $xy = 8$:

The locus of the foot of the perpendicular dropped from the focus $S(ae, 0)$ to any variable tangent line of the hyperbola $\ds\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ matches which curve?

Find the equation of the normal line to the rectangular hyperbola $xy = 4$ at the parameter point $t = 1$:

For what value of the intercept constant $c$ is the straight line $y = 3x + c$ tangent to the hyperbola $\ds\frac{x^2}{4} - \frac{y^2}{9} = 1$?

If a circle cuts a rectangular hyperbola $xy = c^2$ at four points parameterized by $t_1, t_2, t_3,$ and $t_4$, the parameter product $t_1 t_2 t_3 t_4$ is identically:

The acute angle between the two asymptotes of a standard hyperbola whose semi-axes satisfy $b = a$ is exactly:

Find the radius value of the director circle for the hyperbola $\ds\frac{x^2}{16} - \frac{y^2}{7} = 1$.

If the normal line drawn at parameter point $t = 2$ on the rectangular hyperbola $xy = c^2$ meets the curve again at a new parameter point $t'$, find the integer magnitude value of the denominator $k$ if $t' = -\ds\frac{1}{k}$.

Calculate the total number of real normal lines that can be drawn from the geometric center $(0,0)$ to a standard non-equilateral hyperbola.

Assertion (A): The eccentricity of any rectangular hyperbola curve configuration is a fixed scalar constant equal to $\sqrt{2}$.
Reason (R): For a rectangular hyperbola, the lengths of the semi-axes are equal ($a = b$), which simplifies the eccentricity identity to $e^2 = 1 + \ds\frac{a^2}{a^2} = 2$.

Assertion (A): No real director circle can be constructed for the standard hyperbola equation $\ds\frac{x^2}{9} - \frac{y^2}{16} = 1$.
Reason (R): The director circle radius expression matches $\sqrt{a^2 - b^2}$, which evaluates to an imaginary value $\sqrt{9 - 16} = \sqrt{-7}$ when $a < b$.