Central Conicoids, Tangent Plane And Normal

The general Equation of second degree (conicoid) :

The surface represented by the general equation of second degree in x,y,z

is called a conicoid or quadric.

The standard equation of the central conicoid –
The surface represented by the equation ax²+by²+cz²=⊥ (1) possesses the property that all chords of the surface which pass through the origin are bisected at the origin.
The conicoid represented by the equation (1) possesses a unique centre (origin) that is, the origin is the only point which possesses this property. This conicoid is therefore known as the central conicoid and the equation (1) is its standard equation.
The equation (1) represents three different surface depending upon the signs of the coefficients a,b,c. if a,b,c are all positive then the surface is an ellipsoid, if one of them is negative the surface is the hyperboloid of one sheet. In case all the three are negative the surface is imaginary.

Tangent lines and tangent plane

Definition – A straight line which intersects a central conicoid in two coincident point is called a tangent line to the central conicoid at that point.

The locus of all tangent lines at a point on a central conicoid is called a tangent plane to the central conicoid at that point.

Equation of the tangent plane at a point : The equation of the tangent plane to the conicoid

Director sphere :-
Definition- The locus of the point of intersection of three mutually perpendicular tangent planes to a central conicoid is a sphere, concentric with the conicoid, called the Director sphere of the conicoid.
 The locus of the point of intersection of three mutually perpendicular tangent planes to a central conicoid