Second Degree Equations
We will cover following topics
Second Degree Equations In Three Variables
The general equation of the second degree in in three variables is given by f(x,y,z)=ax2+by2+cz2+2fyz+2gxz+2hxy+2px+2qy+2rz+d=0.
This equation describes a quadric surface, also known as a conicoid.
Some examples of conicoids are ellipsoids, hyperboloids and paraboloids.
Reduction To Canonical Forms
The equation for a conicoid can be reduced to its canoninal (normal) form by change of coordinate system. Some examples of conicoids and their canonical forms are given below.
Hyperboloid
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Hyperboloid of one sheet: x2a2+y2b2−z2c2−1=0
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Hyperboloid of two sheets: x2a2+y2b2−z2c2+1=0
PYQS
Second Degree Equations in Three Variables
1) Find the equation of the tangent at the point (1,1,1) to the conicoid 3x2−y2=2z.
[2017, 10M]
2) Find the locus of the points of intersection of three mutually perpendicular tangent planes to ax2+by2+cz2=1.
[2017, 10M]
3) Find the locus of the point of intersection of three mutually perpendicular tangent planes to the conicoid ax2+by2+cz2=1.
[2016, 15M]
4) A line is drawn through a variable point on the ellipse x2a2+y2b2=1, z=0 to meet two fixed lines y=mx, z=c and y=−mx, $z=−c. Find the locus of the line.
[2009, 12M]
5) Find the locus of the point which moves so that its distance from the plane x+y−z=1 is twice its distance from the line x=−y=z.
[2007, 12M]
6) A pair of tangents to the conic ax2+by2=1 intercepts a constant distance 2k on the y−axis. Prove that the locus of their point of intersection is the conic
ax2(ax2+by2−1)=bk2(ax2−1)2[2006, 12M]
7) If PSP′ and QSQ′ are the two perpendicular focal chords of a conic 1r=1+ecosθ, prove that
1SP.SP′+1SQ.SQ′is constant.
[2006, 15M]
8) Find the locus of the middle points of the chords of the rectangular hyperbola x2−y2=a2 which touch the parabola y2=4ax.
[2004, 15M]
9) Prove that the locus of the foot of the perpendicular drawn from the vertex on a tangent to the parabola y2=4ax is (x+a)y2+x3=0.
[2004, 12M]