Cone

• A cone is a surface generated by a moving straight line (generator) that passes through a fixed point (vertex) and interesects a given curve (guiding curve).

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• Any homogeneous equation in \(x\), \(y\), \(z\) represents a cone with its vertex at origin.

  Note: \(f(x,y,z)=0\) is homogeneous if \(f(rx,ry,rz)=0\), \(\forall r\)

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• A cone is called quadric if its equation is of second degree in \(x\), \(y\) and \(z\). The general equation of a quadric cone is \(a x^{2}+b y^{2}+c z^{2}+2 f y z+2 g z x+2 h x y=0\), with vertex as the origin.

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• The equation of the cone with vertex as the origin and which passes through the curve of intersection of the plane \(lx+my+nz=p\) and the surface \(a x^{2}+b y^{2}+c z^{2}=1\) is given by:

\[p^{2}\left(a x^{2}+b y^{2}+c z^{2}\right)=(l x+m y+n z)^{2}\]

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• The equation of the cone with vertex as \((\alpha,\beta,\gamma)\) and base the conic \(a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+e=0, z=0\) is given by: \(a(\alpha z-y x)^{2}\) + \(2 h(\alpha z-y x)(\beta z-v y)\)+ \(b(\beta z-w)^{2}\)+\(2 g(\alpha z-f x)(z-\gamma)\)+\(2 f(\beta z-w)(z-\gamma)\)+\(c(q-z)^{2}=0\)

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• A right circular cone is a cone in which the generator makes a fixed angle with a fixed line passing through vertex.

  The fixed angle is caled the semi-vertical angle and the fixed line is called the axis of the cone.

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• The equation of the right circular cone with vertex as origin, axis the \(z-axis\) and semi-vertical angle \(\alpha\) is \(x^{2}+y^{2}=z^{2} \tan ^{2} \alpha\).

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• The equation of a right circular cone with vertex as the origin, axis the line \(\dfrac{x}{l}=\dfrac{y}{m}=\dfrac{z}{n}\) and semi-vertical angle \(\alpha\) is

\[(l x+m y+n z)^{2}=\left(x^{2}+y^{2}+z^{2}\right) \cos \alpha\]

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• The equation of a right circular cone with vertex as \((\alpha, \beta, \gamma)\), semi vertical angle \(\theta\) and axis having directon cosines \(l\), \(m\) and \(n\) is given by: \([l(x-\alpha)\)+\(m(y-\beta)\)+\(n(z-\gamma)]^{2}\)=\([(x-\alpha)^{2}\)+\((y-\beta)^{2}\)+\((z-y)^{2}] \cos ^{2} \theta\)

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• If the direction cosines of a line satisfy a homogeneous equation, then the line is a generator of a cone.

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• The equation of a cone with \(x\), \(y\), \(z\) axes as generators and vertex as origin is given by \(f y z+g z x+h x y=0\).

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• The equation of the tangent (enveloping) cone from the point \(P(x_1,y_1,z_1)\) to the sphere \(x^{2}\)+\(y^{2}\)+\(z^{2}\)+\(2 u x\)+\(2 v y\)+\(2 w z\)+\(d=0\) is given by:

\[SS_1 = T^2\]

where

\(S=x^{2}+y^{2}+z^{2}+2 u x+2 z y+2 u z+d\),

\(\mathrm{S}_{1}=x_{1}^{2}+y_{1}^{2}+z_{1}^{2}+2 u x_{1}+2 u y_{1}+2 u z_{1}+d\), and

\[T=x x_{1}+y y_{1}+z z_{1}+u\left(x+x_{1}\right)+v\left(y+y_{1}\right)+w\left(z+z_{1}\right)+d\]

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• The condition that the equation \(a x^{2}+b y^{2}+c z^{2}+2 f y z+2 g z x+2 h x y+2 u x+2 v y+2 w z+d=0\) may represent a cone is given by: \(\begin{vmatrix} {a} & {h} & {g} & {u} \\ {h} & {b} & {f} & {v} \\ {g} & {f} & {c} & {w} \\ {u} & {v} & {w} & {d} \end{vmatrix}\)

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• Working Rule to determine the vertex of a cone represented by equation \(f(x,y,z)=0\):

(i) Introduce powers of \(t\) in the equation and make it homogeneous in \(x\), \(y\), \(z\) and \(t\)

(ii) Find partial derivatives \(\dfrac{\partial \mathrm{F}}{\partial x}\), \(\dfrac{\partial \mathrm{F}}{\partial y}\), \(\dfrac{\partial \mathrm{F}}{\partial z}\) and \(\dfrac{\partial \mathrm{F}}{\partial t}\)

(iii) Substitute \(t=1\) in above four partial derivatives and equate to 0.

(iv) Solve any three equations and this solution is the vertex of the cone.

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• The equation of the tangent plane at the point \((x_1, y_1, z_1)\) of the cone \(a x^{2}+b y^{2}+c z^{2}+2 f y z+2 g z x+2 h x y=0\) is given by:

\[x\left(a x_{1}+h y_{1}+g z_{1}\right)+y\left(h x_{1}+b y_{1}+f z_{1}\right)+z\left(g x_{1}+f_{1}+c z_{1}\right)=0\]

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• The condition that the plane \(l x+m y+n z=0\) may touch the cone \(a x^{2}+b y^{2}+c z^{2}+2 f y z+2 g z x+2 h x y=0\) is given by:

\[\mathrm{A} l^{2}+\mathrm{Bm}^{2}+\mathrm{Cn}^{2}+2 \mathrm{F}mn+2 \mathrm{G} n l+2 \mathrm{H}lm=0\]

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• The condition that the cone \(a x^{2}+b y^{2}+c z^{2}+2 f y z+2 g z x+2 h x y=0\) may have three mutually perperdicular generators is given by: \(a+b+c=0\)

Cone

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