Assertion Reasoning Quiz

Assertion (A): A subatomic particle follows a hyperbolic orbital path. If the linear asymptotes of this specific path intersect exactly at a \(90^\circ\) angle, the eccentricity of the particle's orbit is strictly \(\sqrt{2}\).
Reason (R): Asymptotes intersecting at \(90^\circ\) dictate that \(\tan(45^\circ) = \frac{b}{a} = 1\). This implies \(a=b\), rendering the path an equilateral hyperbola with \(e = \frac{\sqrt{a^2+a^2}}{a} = \sqrt{2}\).

Assertion (A): A point moves such that the sum of the squares of its distances from \((2, 0)\) and \((-2, 0)\) is always \(16\). The locus of this moving point is a circle of radius \(2\).
Reason (R): The geometric condition algebraically simplifies to \(x^2 + y^2 = 16\), defining a circle of radius \(4\).

Assertion (A): The length of the latus rectum of the ellipse \(16x^2 + y^2 = 16\) is \(0.5\).
Reason (R): The foci of this specific ellipse are mathematically located exclusively on the y-axis.

Assertion (A): A marine radar located at the origin with a maximum scanning radius of \(10\) km will successfully detect a stationary ship located at the coordinates \((6, 8)\).
Reason (R): The ship's distance from the radar is exactly \(10\) km, meaning it lies securely on the outermost boundary of the radar's coverage area.

Assertion (A): If a geometric plane seamlessly intersects a double-napped cone exactly at its central vertex, the resulting conic section is exclusively and always a single point.
Reason (R): When the intersecting plane forces its way through the vertex and the angle it makes with the vertical axis (\(\beta\)) is strictly greater than the cone's intrinsic semi-vertical angle (\(\alpha\)), the section completely degenerates into a solitary point.

Assertion (A): The specific coordinate point \((4, 5)\) does not rest anywhere on the geometric boundary of the hyperbola \(\frac{x^2}{16} - \frac{y^2}{9} = 1\).
Reason (R): A point \((x_1, y_1)\) can mathematically satisfy a hyperbola's equation only if its coordinates are strictly positive, primarily dictating \(x_1 > a\) and \(y_1 > b\).