If a and c are positive real numbers and the | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The equation of the ellipse is $\frac{x^2}{(2c)^2} + \frac{y^2}{c^2} = 1$. The vertices are at $(\pm 2c, 0)$ and $(0, \pm c)$.The equation of the

Vedclass · Accepted Answer

$9ac - 9a^2 - 2c^2 > 0$

Vedclass · Answer

(C) The equation of the ellipse is $\frac{x^2}{(2c)^2} + \frac{y^2}{c^2} = 1$. The vertices are at $(\pm 2c, 0)$ and $(0, \pm c)$.The equation of the circle is $x^2 + y^2 = (3a)^2$. The radius is $3a$.For the circle and ellipse to have four distinct points of intersection,the radius of the circle must be greater than the semi-minor axis of the ellipse and less than the semi-major axis of the ellipse.Thus,$c From $3a From $c  0$.Substituting $y^2 = 9a^2 - x^2$ into the ellipse equation: $\frac{x^2}{4c^2} + \frac{9a^2 - x^2}{c^2} = 1$.$x^2 + 4(9a^2 - x^2) = 4c^2$ $\Rightarrow x^2 + 36a^2 - 4x^2 = 4c^2$ $\Rightarrow 3x^2 = 36a^2 - 4c^2$ $\Rightarrow x^2 = 12a^2 - \frac{4}{3}c^2$.For two distinct values of $x^2$ (which give four points),we need $0 $0 $12a^2 - \frac{4}{3}c^2 > 0$ $\Rightarrow 36a^2 > 4c^2$ $\Rightarrow 9a^2 > c^2$ $\Rightarrow 3a > c$.$12a^2 - \frac{4}{3}c^2 Combining these,$c  0$ is derived from the intersection analysis.

If $a$ and $c$ are positive real numbers and the ellipse $\frac{x^2}{4c^2} + \frac{y^2}{c^2} = 1$ has four distinct points in common with the circle $x^2 + y^2 = 9a^2$,then

Similar Questions

The length of the latus rectum of an ellipse is $6$ units and the distance between a focus and its nearest vertex on the major axis is $\frac{5}{3}$ units. If $e$ is the eccentricity of this ellipse,then $e$ satisfies the equation:

Find the equation of the ellipse whose latus rectum is $10$ and the length of the minor axis is equal to the distance between the foci.

The area (in sq. units) of the triangle formed by the tangent and normal at a point $\left(\frac{a}{\sqrt{2}}, \frac{b}{\sqrt{2}}\right)$ to the curve $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and the $X$-axis is

The number of real tangents that can be drawn to the ellipse $3x^2 + 5y^2 = 32$ passing through $(3, 5)$ is

The eccentricity of the ellipse $25x^2 + 16y^2 = 100$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module