Ellipse practice · Mixed difficulty

Option A (10) is correct because the sum of the focal distances of any point on the ellipse is equal to the length of the major axis, which is $2a = 10$.

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 (${b^2}:{a^2}$) is correct because it represents the ratio of PR to RS, derived from the equation of the normal to the ellipse and the coordinates of the points where the normal intersects the axes.

Option C (\left(1-e²\right): 1) is correct because it represents the ratio of PT to PG, which is equal to b²/a², and b²/a² = 1-e² for an ellipse.

The correct answer is $\frac{\pi }{2}$ because the locus of the point of intersection of tangents to the second ellipse is the director circle, implying a right angle between the tangents.

Option C ($13 / 5$) is correct because the eccentricities are equated and the ratio $a/b$ is calculated.

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 D ($16: 9$) is incorrect. Option C ($9: 16$) is correct because it represents the ratio of $P G_1$ and $P G_2$. The correct ratio is derived from the geometry of the ellipse and its normal.

Option A ($25x^2 + 16y^2 = (x^2 + y^2)^2$) is correct because it represents the locus of the foot of the perpendicular from the center to any tangent of the ellipse.

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 C ($K \in ( - 3,2)$) is correct because for the major axis to coincide with the x-axis, the coefficient of x² must be greater than the coefficient of y², leading to the inequality $K^2 + 2K + 5 < K + 11$.

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 A ($\left(\frac{a^2-b^2}{a^2+b^2}\right) a b$) is correct because the area of the rectangle is the product of the perpendicular distances from the center to the tangent and normal, which simplifies to the given expression.

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 D ($\frac{\left(a^2-b^2\right)^2 m^2}{a^2 m^2+b^2}$) is correct because it provides the correct condition for the line to be a 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.

Option A ($b^2 / a$) is correct because the harmonic mean of the segments of a focal chord is equal to the semi-latus rectum.

Option A ($\frac{x^2}{25}+\frac{y^2}{21}=1$) is correct because it satisfies the given conditions for the ellipse's major axis and foci.

Option B ($\frac{{\pi}}{{\sqrt{{2}}}}$) is correct because after finding the length of the latus rectum of ellipse E1 and equating it to the length of the latus rectum of ellipse E2, we can find p and then the area of ellipse E2.

Option B ($\left( { - \frac{2}{{\sqrt {11} }},\frac{1}{{\sqrt {11} }}} \right)$ or $\left( {\frac{2}{{\sqrt {11} }}, - \frac{1}{{\sqrt {11} }}} \right)$) is correct because it represents the points on the ellipse where the tangent's slope matches the given line's slope.

Option A (2) is correct because the area of the triangle formed by the points on the ellipse is Kabsin((φ-ψ)/2)sin((ψ-θ)/2)sin((θ-φ)/2), where K equals 2.

Option A ($2 a$) is correct because the sum of the focal distances of any point on the ellipse is equal to the length of the major axis, which is $2a$.

Option D (-12) is correct because by comparing the given normal equation to the standard form, the value of k is found to be -12.

Option A ($7 x^2+2 x y+7 y^2+10 x-10 y+7=0$) is correct because it represents the equation of the ellipse derived using the focus, directrix, and eccentricity.

Option C (1, 2/3) is correct because it represents the lengths of the major and minor axes after converting the given equation to standard ellipse form.

Option A ($\frac{{\sqrt{{41}}}}{{5}}$) is correct because it is the length of AB calculated using the equation of the chord with the given midpoint.