Hyperbola practice · Mixed difficulty

Option A ($b^2$) is correct because the product of the lengths of the perpendiculars from the foci to any tangent of the hyperbola is equal to $b^2$.

Option A ($2 / \sqrt{3}$) is correct because it is the eccentricity of the hyperbola conjugate to the given hyperbola, calculated using the relationship between the eccentricities of conjugate hyperbolas.

Option B ($\frac{5}{4}$) is correct because it is the eccentricity calculated from the given latus rectum of the hyperbola.

Option B (ab) is correct because it represents the area of the triangle formed by the asymptotes and the tangent at the vertex of the hyperbola.

Option A (A hyperbola) is correct because the equation simplifies to xy = ab, which represents a hyperbola.

Option A ($\frac{1}{2 \sqrt{2}}$) is correct because it is the limit of $n \cdot d_n$ as $n$ approaches infinity, calculated using algebraic manipulation and limit properties.

Option D (-24) is correct. The condition for tangency to the hyperbola gives the value of k, and then the condition for tangency to the parabola gives the value of α.

Option B ($4 x+3 y+17=0$) is correct because the asymptotes of a rectangular hyperbola are perpendicular, and the center lies on the given line. By finding the center and using the perpendicularity condition, we can determine the equation of the other asymptote.

Option C ($24 \sqrt{\frac{{17}}{{20}}}$) is correct because it involves finding the slope of the tangent to the hyperbola, then using that slope to find 'a' for the parabola and finally calculating the latus rectum.

Option C (Hyperbola) is correct because the equation derived by eliminating the parameter $m$ results in the standard form of a hyperbola.

Option A (0) is correct because the sum of the y-coordinates of the intersection points is zero, as derived from the equation of the circle and hyperbola.

Option B ($t^3 t^{\prime}=-1$) is correct because it is derived from the equation of the normal to the hyperbola and the condition that it intersects the curve again at a different point.

Option C $2\left(\tan ^{{-1}} \frac{{1}}{{3}}+\tan ^{{-1}} \frac{{1}}{{2}}\right)$ is correct because the point lies on the director circle, implying a right angle between tangents, and the expression simplifies to $\frac{{\pi}}{{2}}$.

Option C ($a\left( {e + \cos \alpha } \right)$) is correct because it represents the distance between the point T on the transverse axis and a focus of the hyperbola.

Option A (10) is correct because the eccentricity of the conjugate hyperbola is calculated, and then 8e is found to be 10.

Option C ($\lambda < -2$ or $\lambda > 2$) is correct because the eccentricity of a hyperbola is greater than 1, leading to the condition $|\lambda| > 2$.

Option A ($\sqrt {21} \left( {x - 2y} \right) = 41$) is correct because it represents the equation of the normal to the hyperbola that is perpendicular to the given line, found by using trigonometric substitution and the condition for perpendicularity.

Option C (8) is correct because after rewriting the equation in standard form, the distance between the foci is calculated as 2ae, which equals 8.

Option C ($4 x^2-3 y^2-36=0$) is correct because it represents a hyperbola with vertices at $(\pm 3, 0)$ and the given semi-latus rectum.

Option C ($y=3x+5$ and $y=3x-5$) is correct because by finding the slope of the given line and using the tangent condition for a hyperbola, we can determine the equations of the tangents parallel to the line.

Option B (an ellipse whose eccentricity is $\sqrt{{\frac{{2}}{{r+1}}}}, \text{{ when }} r > 1$) is correct because when $r > 1$, the equation represents an ellipse, and the eccentricity is calculated correctly.

Option A ($\frac{{8\sqrt 2 }}{{\sqrt 3 }}$) is correct because it represents the length of the transverse axis of the given hyperbola after rewriting the equation in standard form.

Option B (3/2) is correct because after converting the equation to standard form, the eccentricity can be calculated using the relationship between a, b, and e.

Option A ($\pi - 2{\tan ^{ - 1}}\frac{3}{4}$) is correct because it is derived from the formula for the angle between the asymptotes of a hyperbola.

Option A ($\frac{\pi }{2}$) is correct because the point lies on the director circle of the hyperbola, implying the tangents are perpendicular.

Option B ($9{x^2} - 8{y^2} - 18x + 9 = 0$) is correct because it is derived from the condition that x=9 is the chord of contact and using the equation $T^2 = SS_1$.

Option A (5) is correct because the product CQ * CR is calculated using the tangent equation and the given hyperbola.

Option B ($3x - 4y = 0$) is correct because the locus of the midpoints of chords parallel to a given line can be found using the equation of the chord with midpoint (h,k) and equating its slope to the slope of the given line.

Option A (8) is correct because the difference in the focal distances of any point on a hyperbola is equal to the length of the transverse axis, which is given as 8.

Option D ($x = \frac{9}{{\sqrt {13} }}$) is correct because the directrix of the hyperbola is calculated using the formula $x = \pm \frac{a}{e}$.