Ellipse practice · Mixed difficulty

Option B ($\frac{1}{\sqrt{2}}$) is correct because the eccentricity is calculated using the relationship between the latus rectum and the major axis.

Option B ($\frac{{10}}{3}$) is correct because the length of the latus rectum of an ellipse is given by $\frac{{2{b^2}}}{a}$, where a and b are the semi-major and semi-minor axes respectively.

Option C ($\frac{\sqrt{2}}{3}$) is correct because it is derived from the equation of the normal to the ellipse and the given x-intercept, combined with the ellipse equation to solve for the y-coordinate of point P.

Option C ($\left(\frac{2}{\sqrt{10}}, \frac{9}{\sqrt{10}}\right)$) is correct because it represents a point on the ellipse where the normal's slope matches that of the given line.

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 D is correct because the mean position is derived from the properties of normals to the ellipse and the equation of the ellipse itself.

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 D ($x^2+y^2=1$) is correct because the equation of the circle passing through the intersection points of the two ellipses is found by setting up a general equation and solving for the parameter that makes it a circle.

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 D ($\frac{\pi}{2}-3 \theta$) is correct because it is derived from the property that the sum of eccentric angles where a circle cuts an ellipse is an even multiple of $\pi$.

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 C (P lies inside C but outside E) is correct because when the coordinates of P are substituted into the equations of the ellipse and the circle, it satisfies the conditions.

Option C ($25 x^2+25 y^2-150 x-31=0$) is correct because the center of the circle is at the focus of the ellipse (3,0) and the radius is the semi-latus rectum (16/5).

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 A (Major axis) is correct because the sum of the focal distances of any point on an ellipse is constant and equal to the length of the major axis (2a).