Ellipse practice · Easy

Option A ($\frac{13}{36}$) is correct because it represents the value of $CA^{-2} + CB^{-2}$ derived from the given conic equation and the condition $\angle ACB = \pi/2$.

Option A ($4 a^2 x^2-b^2 y^2=4 a^4$) is correct because the polar of the parabola touches the ellipse, so the condition for tangency can be used to find the locus of the pole.

Option A ($\frac{{\left(a^2-b^2\right) a b}}{{a^2+b^2}}$) is correct because it is the product of the perpendicular distances from the center to the tangent and normal at the given point.

Option C ($a^2 \cos ^2 \alpha+b^2 \sin ^2 \alpha=p^2$) is correct because it is the condition for the line to be tangent to the ellipse, derived by comparing the given line to the standard form and using the tangency condition.

Option A (an ellipse) is correct because the parametric equations can be rewritten into the standard form of an ellipse equation.

Option B ($\sqrt {\frac{2}{3}}$) is correct because it is derived from the given relationship between the latus rectum and the major axis of the ellipse.

Option B ($\frac{{\sqrt{{5}}+1}}{{2}}$) is correct because it represents the maximum distance from the center to any point on the ellipse, derived by maximizing the distance function.

Option B (63) is correct because it satisfies the condition for the two ellipses to have the same eccentricity.

Option C ($\tan ^{-1}\left( \pm \frac{b}{a e}\right)$) is correct because the eccentric angle is found by equating the coordinates of a general point on the ellipse with the coordinates of the latus rectum's end points.

Option B (${x^2} + {y^2} = 41$) is correct because the locus of the intersection of perpendicular tangents to an ellipse is its director circle, given by ${x^2} + {y^2} = {a^2} + {b^2}$.

Option D ($\pm \sqrt{132}$) is correct because the condition for tangency of a line to an ellipse is $c^2 = a^2m^2 + b^2$, where $c$ is the y-intercept, $m$ is the slope, and $a$ and $b$ are the semi-major and semi-minor axes.

Option A ($3/4$) is correct because the eccentricity is derived from the geometric condition that the lines joining a focus to the extremities of the minor axis form an equilateral triangle.

Option B ((√5 - 1)/2) is correct because the eccentricity is derived from the relationship between the distance between the foci and the length of the latus rectum.

Option B is correct because the equation of the ellipse is derived from the given focus and eccentricity, resulting in \frac{{{x²}}}{{{{5²}}}}} + \frac{{{y²}}}{{{{3²}}}}} = 1.

Option A (y=3, x+y=5) is correct because these are the equations of the tangents to the ellipse that pass through the point (2,3), found by using the tangency condition.

Option C ($\frac{\sqrt{26}}{5+\sqrt{29}}$) is correct because it is the eccentricity of the ellipse passing through the origin with the given foci.

Option B ($x+4y=10$) is correct because it is the equation of the chord of the ellipse with the given midpoint (2, 2).

Option C ($8 / 3$) is correct because using the given focus, directrix, and eccentricity in the formula relating them gives $a = 8/3$.

Option D ($3 x^2+5 y^2-32=0$) is correct because it satisfies both the point (-3,1) and the eccentricity condition.

Option B ($y = 3x \pm \sqrt {\frac{{155}}{{12}}}$) is correct because it represents the equations of the tangents to the ellipse that are parallel to the given line, calculated using the tangent's slope.

Option C (10) is correct because the sum of the distances from any point on the ellipse to the foci is equal to the length of the major axis.

Option A ($( \pm 3,0)$) is correct because after converting the equation to standard form, the foci are calculated as $( \pm ae, 0)$.

Option C ($\frac{1}{{\sqrt 3 }}$) is correct because the eccentricity is derived using the slopes of conjugate diameters and the relationship between a, b, and e in an ellipse.

Option B ($\frac{{(x-2)^2}}{4}+\frac{{(y+3)^2}}{3}=1$) is correct because it is derived from the given center, focus, and vertex to form the equation of the ellipse.

Option D (Centre and vertices only) is correct because the ellipse and hyperbola share the same center (0,0) and vertices (+-5,0), but different foci and directrices.

Option A ($a^2 l^2+b^2 m^2=n^2$) is correct because it represents the condition for tangency of a line to an ellipse, derived by substituting the line equation into the ellipse equation and applying the discriminant condition for a single solution.

Option A (x²=a(a-y)) is correct because it represents the locus of the extremities of the latus recta of the given ellipse.

Option C ($\frac{{25}}{{12}}$) is correct because it is the sum of the eccentricities of the ellipse and hyperbola, calculated using the given equations.

Option D ($\pm \frac{{\sqrt 3 }}{2}$) is correct because these are the values of $\lambda$ for which the given line is normal to the ellipse.

Option B (3) is correct because using the property definition of the ellipse, the eccentricity 'e' is found to be 1/3, and 9e equals 3.