Find the eccentricity of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ whose latus rectum is half of its major axis. $\left(a>b\right)$

Find the equation of the parabola that satisfies the following conditions: Vertex $(0,0)$, focus $(-2,0)$

Tangents are drawn to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at two points whose eccentric angles are $\alpha -\beta$ and $\alpha +\beta$ The coordinates of their point of intersection are

The tangent at a point $P\left(a\cos \phi ,b\sin \phi \right)$ of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets its auxiliary circle at two points, the chord joining which subtends a right angle at the center. Find the eccentricity of the ellipse.

If the normals to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at the points $\left(X_1,y_1\right),\left(x_2,y_2\right)\text{and}\left(x_3,y_3\right)$ are concurrent, prove that $\left|\begin{array}{ccc}{x}_1& y_1& x_1{y}_1\\ x_2& y_2& x_2{y}_2\\ x_3& y_3& x_3{y}_3\end{array}\right|=0$.

Find the area of the greatest rectangle that can be inscribed in an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Find the centre and radius of the circle $x^2+y^2-8x+10y-12=0$

Find the equation of the hyperbola satisfying the give conditions: Vertices $\left(\pm 7,0\right)$
$e=\frac{4}{3}$