Conic Sections
Consider a branch of the hyperbola \[x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y-3=0\] with vertex at a point \(A.\) Also, let \(B\) be one of the end points of its latus rectum. If \(C\) is the focus of the hyperbola nearest to point \(A,\) what is the area of triangle \(ABC ?\)
What are the coordinates of the foci of the rectangular hyperbola \[x^2-y^2=7 ^2?\]
What are the coordinates of the vertices of the hyperbola \[4x^2-9y^2-40x+54y -17=0 ?\]
Given points \(F=(5,0)\) and \(F'=(-5,0),\) what is the equation of the locus of points \(P\) in a plane such that \[ \left\lvert \left\lvert \overline{PF} \right\rvert - \left\lvert \overline{PF'} \right\rvert \right\rvert= 8 ?\]
Note: \(\lvert \overline{AB} \rvert\) denotes the length of line segment \(\overline{AB}.\)
Suppose that \((7,12)\) and \((11,8)\) are the foci of a hyperbola passing through the origin. Then the eccentricity of the hyperbola is \(\text{_________}.\)