HPAS Maths Optional PYQs: Analytical Geometry
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On that page, you will find year-wise and question-wise solutions.
The sources provide numerous questions from the HPAS Maths Optional Question Paper-1 examinations related to Analytical Geometry, which includes 2D Conics, 3D Geometry (Planes, Lines, Spheres, Cylinders, Cones), and Conicoids (General Second Degree Equation in 3D). Here are the year-wise questions categorized under Analytical Geometry:
HPAS 2024 Analytical Geometry Questions
Question 3(b): Find the equation of the plane passing through the line of intersection of the planes \(a_1x+b_1y+c_1z+d_1=0\) and \(a_2x+b_2y+c_2z+d_2=0\) and perpendicular to the xy-plane.
HPAS 2023 Analytical Geometry Questions
Question 2(c): Find the equations of the lines in which the plane \(2x+y-z=0\) cuts the cone \(4x^2 – y^2 + 3z^2 = 0\).
Question 4(b): Find the magnitude and the equations of the shortest distance between the lines: \(\frac{x}{2} = \frac{-y}{3} = \frac{z}{1}\) and \(\frac{x-2}{3} = \frac{y-1}{-5} = \frac{z+2}{2}\).
HPAS 2021 Analytical Geometry Questions
Question 1(c): Show that the tangent planes at the extremities of any diameter of an ellipsoid are parallel.
Question 3(a): Show that the equation \(ax^2+by^2+cz^2+2ux+2vy+2wz+d=0\) represents a cone if: \(\frac{u^2}{a} + \frac{v^2}{b} + \frac{w^2}{c} = d\).
Question 7(a): Determine the center and radius of the circle in which the sphere \(x^2+y^2+z^2+2x-2y-4z-19=0\) is cut by the plane \(x+2y+2z+7=0\).
HPAS 2020 Analytical Geometry Questions
Question 3(a): Find the locus of the point of intersection of three mutually perpendicular tangent planes to \(ax^2+by^2+cz^2=1\).
Question 4(c): Reduce the following equation to the standard form: \(3x^2+5y^2+3z^2+2yz+2zx+2xy-4x-8z+5=0\). Find the nature of the conicoid, its center, and the equations of its axes.
HPAS 2018 Analytical Geometry Questions
Question 3(b): If P, Q, and R are three points having Cartesian coordinates (3, -2, -1), (1, 3, 4), and (2, 1, -2) respectively, find the distance from point P to the plane OQR, where O is the origin.
HPAS 2017 Analytical Geometry Questions
Question 4(a): Find the condition that the straight line \(\frac{l}{r} = A\cos\theta + B\sin\theta\) may touch the circle \(r = 2a\cos\theta\) (in polar coordinates).
Question 4(b): Find the equation of a sphere which passes through the points (1, 0, 0), (0, 1, 0), and (0, 0, 1) and has the smallest possible radius.
Question 7(b): Two forces act, one along the line \(y=0, z=0\) and the other along the line \(x=0, z=c\). Show that the surface generated by the central axis of their equivalent wrench is \((x^2+y^2)z = cy^2\).
HPAS 2016 Analytical Geometry Questions
Question 7(a): Show that the equation: \(2x^2 – 6y^2 – 12z^2 + 18yz + 2zx + xy = 0\) represents a pair of planes and find the angle between them.
Question 7(b): Find the equations of the tangent planes to the hyperboloid \(2x^2 – 6y^2 + 3z^2 = 5\) which pass through the lines \(3x-3y+6z-5=0\) and \(x+9y-3z=0\).
HPAS 2015 Analytical Geometry Questions
Question 5(a): Reduce the equation \(11y^{2}+14yz+8zx+14xy-6x-16y+2z-2=0\) to canonical form and state the nature of the surface (Conicoid).
Question 5(b): Find the equation of the sphere having the circle: \(x^{2}+y^{2}+z^{2}+10y-4z=8, x+y+z=3\) as a great circle.
Question 5(c): Find the equation of the cylinder whose generators are parallel to the line: \(\frac{x}{1}=\frac{y}{-2}=\frac{z}{3}\) and whose guiding curve is the ellipse: \(x^2 + 2y^2 = 1, z=0\).
HPAS 2014 Analytical Geometry Questions
Question 1(d): Find the surface represented by the equation: \(x^{2}+4y^{2}+z^{2}-4yz+2zx-4xy -2x+4y-2z-3=0\).
Question 5(b): What conic does the equation: \(13x^{2}-18xy+37y^{2}+2x+14y-2=0\) represent? Find its centre and the equation to the conic referred to the centre as origin (2D Conics).
Question 5(c): Find length and the equations to the shortest distance between the following lines: \(\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}\) and \(\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}\).
To solidify the understanding of these topics, particularly the 3D analytical geometry component, one can think of the general second-degree equation (like the one in Question 1(d) of 2014 or Question 5(a) of 2015) as an unidentified structure in space. The goal of analytical geometry is like giving a blueprint to that structure: determining if it’s a house (ellipsoid), a tower (cylinder), a funnel (cone), or perhaps just two flat walls (pair of planes), and then mathematically standardizing the description to simplify its orientation and position.