10 Question MCQ (including Assertion)

Assertion (A): A man running notes that the sum of his distances from two static flag posts is always \(10\) m, and the distance between the posts is \(8\) m. His path forms an ellipse with an eccentricity of \(4/5\).
Reason (R): The area of the largest rectangle that can be successfully inscribed inside an ellipse defined by \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) is geometrically \(2ab\).

If the distance between the foci of an ellipse is equal to the length of its latus rectum, find its eccentricity \(e\).

Assertion (A): A \(15\) cm rod rests with its endpoints freely sliding on the x and y axes. A point \(P\) marked \(6\) cm from the end touching the x-axis traces an elliptical locus in the plane.
Reason (R): The exact equation of the locus traced by point \(P\) evaluates to \(\frac{x^2}{36} + \frac{y^2}{81} = 1\).

Find the coordinates of the focus and the equation of the directrix for the parabola \(y^2 = -8x\).

Assertion (A): The equation of a circle passing through the origin and making intercepts \(a\) and \(b\) on the positive coordinate axes is \(x^2 + y^2 - ax - by = 0\).
Reason (R): The center of such a circle is fixed at the coordinates \((a, b)\).

Find the length of the side of an equilateral triangle inscribed in the parabola \(y^2 = 4ax\), with one vertex at the vertex of the parabola.

A particle follows a hyperbolic path around a massive object located at its focus. If the asymptotes of this path are at an angle of \(90^\circ\) to each other, what is the eccentricity of the particle's orbit?

What is the length of the latus rectum for the hyperbola \(16x^2 - 9y^2 = 576\)?

Assertion (A): The hyperbola strictly defined by \(9y^2 - 4x^2 = 36\) has its transverse axis located entirely along the y-axis.
Reason (R): In the standard layout of a hyperbola, the mathematical variable possessing the positive coefficient designates the specific axis upon which the transverse axis and foci reside.