Analytic Geometry

UPSC Mathematics Optional · Paper I · Conics, 3D Geometry & Conicoids

1. Conics — General Theory 2. Parabola 3. Ellipse 4. Hyperbola 5. Planes & Lines in 3D 6. Sphere 7. Cone & Cylinder 8. Conicoids

§1 · Conics — General Second Degree Curve

General Conic PYQ★★★

\(ax^2+2hxy+by^2+2gx+2fy+c=0\). Discriminant \(\Delta = \begin{vmatrix}a&h&g\\h&b&f\\g&f&c\end{vmatrix}\)

\(\Delta=0\): degenerate (pair of lines or point or empty).
\(\Delta\ne 0\): proper conic.

Classification by \(h^2-ab\) PYQ★★★

\(h^2-ab < 0\): Ellipse (circle if \(a=b\) and \(h=0\)).
\(h^2-ab = 0\): Parabola.
\(h^2-ab > 0\): Hyperbola (rectangular if \(a+b=0\)).

Removing \(xy\) term — Rotation of Axes

Rotate by angle \(\theta\) where \(\cot 2\theta = \dfrac{a-b}{2h}\).

New equation has no \(xy\) term. Principal axes = eigenvector directions of \(\begin{pmatrix}a&h\\h&b\end{pmatrix}\).

§2 · Parabola

Standard Parabola \(y^2 = 4ax\) PYQ★★★

Focus: \((a,0)\). Directrix: \(x=-a\). Vertex: \((0,0)\). Axis: \(y=0\). Latus rectum: \(x=a\), length \(4a\).
Parametric: \((at^2, 2at)\).
Tangent at \((x_1,y_1)\): \(yy_1 = 2a(x+x_1)\).
Tangent in slope form: \(y=mx+\dfrac{a}{m}\), contact point: \(\left(\dfrac{a}{m^2},\dfrac{2a}{m}\right)\).
Normal at \((at^2,2at)\): \(y+tx = 2at+at^3\).
Chord of contact from \((x_1,y_1)\): \(yy_1 = 2a(x+x_1)\).

Focal Chord Property

If \((at_1^2,2at_1)\) and \((at_2^2,2at_2)\) are endpoints of a focal chord, then \(t_1 t_2 = -1\).

Foot of normal from a point on the parabola: 3 normals in general (cubic in \(t\)).

§3 · Ellipse

Standard Ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\), \(a>b\) PYQ★★★

Eccentricity: \(e=\dfrac{c}{a}\), \(c^2=a^2-b^2\), \(0
Foci: \((\pm ae, 0)\). Directrices: \(x=\pm\dfrac{a}{e}\).
Latus rectum: \(x=\pm ae\), length \(\dfrac{2b^2}{a}\).
Parametric: \((a\cos\theta, b\sin\theta)\).
Tangent at \((x_1,y_1)\): \(\dfrac{xx_1}{a^2}+\dfrac{yy_1}{b^2}=1\).
Tangent slope form: \(y=mx\pm\sqrt{a^2m^2+b^2}\).
Normal at \((x_1,y_1)\): \(\dfrac{a^2 x}{x_1}-\dfrac{b^2 y}{y_1}=a^2-b^2\).
Focal distances: \(r_1+r_2=2a\) (sum constant).
Director circle: \(x^2+y^2=a^2+b^2\) (tangents at right angles meet here).

Auxiliary Circle & Eccentric Angle

Auxiliary circle: \(x^2+y^2=a^2\). If \(P=(a\cos\phi,b\sin\phi)\) on ellipse, \(\phi\) = eccentric angle. \(Q=(a\cos\phi,a\sin\phi)\) on auxiliary circle. \(NQ\perp\) major axis.

§4 · Hyperbola

Standard Hyperbola \(\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\) PYQ★★

\(e=\dfrac{c}{a}>1\), \(c^2=a^2+b^2\).
Foci: \((\pm ae,0)\). Directrices: \(x=\pm\dfrac{a}{e}\).
Asymptotes: \(bx\pm ay=0\), i.e. \(y=\pm\dfrac{b}{a}x\).
Parametric: \((a\sec\theta, b\tan\theta)\) or \((a\cosh t, b\sinh t)\).
Tangent at \((x_1,y_1)\): \(\dfrac{xx_1}{a^2}-\dfrac{yy_1}{b^2}=1\).
Focal distance difference: \(|r_1-r_2|=2a\).
Rectangular hyperbola \(xy=c^2\): parametric \((ct, c/t)\), asymptotes \(x=0,y=0\).

Rectangular Hyperbola \(xy=c^2\) PYQ★★

Tangent at \((ct,c/t)\): \(x+t^2y=2ct\).
Normal at \((ct,c/t)\): \(t^3x-ty=c(t^4-1)\).
If \((ct_i,c/t_i)\) are four concyclic points, then \(t_1t_2t_3t_4=1\).

§5 · Planes & Lines in 3D

Plane Equations PYQ★★★

General: \(ax+by+cz+d=0\), normal \((a,b,c)\).
Through \((x_0,y_0,z_0)\) with normal \(\mathbf{n}\): \(\mathbf{n}\cdot(\mathbf{r}-\mathbf{r}_0)=0\).
Intercept form: \(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1\).
Distance from point \((x_1,y_1,z_1)\): \(d=\dfrac{|ax_1+by_1+cz_1+d|}{\sqrt{a^2+b^2+c^2}}\).
Angle between planes: \(\cos\theta = \dfrac{|a_1a_2+b_1b_2+c_1c_2|}{\sqrt{\sum a_i^2}\sqrt{\sum b_i^2}}\).
Plane through 3 points: use determinant form.

Line in 3D PYQ★★★

Symmetric form: \(\dfrac{x-x_0}{l}=\dfrac{y-y_0}{m}=\dfrac{z-z_0}{n}=t\).
Direction cosines: \(l^2+m^2+n^2=1\).
Angle between lines: \(\cos\theta = |l_1l_2+m_1m_2+n_1n_2|\).
Skew lines — shortest distance: \(d = \dfrac{|(\mathbf{b}_1-\mathbf{b}_2)\cdot(\mathbf{d}_1\times\mathbf{d}_2)|}{|\mathbf{d}_1\times\mathbf{d}_2|}\)
Coplanarity of two lines: \((\mathbf{b}_1-\mathbf{b}_2)\cdot(\mathbf{d}_1\times\mathbf{d}_2)=0\).

§6 · Sphere

Sphere Equations PYQ★★

General: \(x^2+y^2+z^2+2ux+2vy+2wz+d=0\). Centre: \((-u,-v,-w)\). Radius: \(\sqrt{u^2+v^2+w^2-d}\).

Tangent plane at \((x_1,y_1,z_1)\): \(xx_1+yy_1+zz_1+u(x+x_1)+v(y+y_1)+w(z+z_1)+d=0\).

Two spheres: radical plane = locus of equal power = \(S_1-S_2=0\).

§7 · Cone & Cylinder

Cone with Vertex at Origin PYQ★★★

Equation is homogeneous of degree 2 in \(x,y,z\): \(ax^2+by^2+cz^2+2fyz+2gzx+2hxy=0\).

Every line through origin lying on the cone is a generator.

Right circular cone, vertex \((0,0,0)\), axis along \(z\), semi-vertical angle \(\alpha\): \(x^2+y^2=z^2\tan^2\alpha\).

Enveloping Cone & Cylinder PYQ★★

Enveloping cone of sphere \(S=0\) from point \(P\): \(SS_1 = T^2\) where \(S_1=S(P)\), \(T\)=tangent condition.

Right circular cylinder, axis \(\dfrac{x-\alpha}{l}=\dfrac{y-\beta}{m}=\dfrac{z-\gamma}{n}\), radius \(r\):

Distance from \((x,y,z)\) to axis = \(r\), using perpendicular distance formula.

§8 · Conicoids (Quadric Surfaces)

Standard Conicoids PYQ★★★

Ellipsoid: \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1\) — bounded, all intercepts real.
Hyperboloid of one sheet: \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}-\dfrac{z^2}{c^2}=1\) — connected, ruled surface.
Hyperboloid of two sheets: \(\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}-\dfrac{z^2}{c^2}=1\) — two disconnected pieces.
Elliptic paraboloid: \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=\dfrac{z}{c}\) — opens one side.
Hyperbolic paraboloid (saddle): \(\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=\dfrac{z}{c}\) — saddle shaped, ruled.
Cone: \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}-\dfrac{z^2}{c^2}=0\).

Tangent Plane to Conicoid PYQ★★

For \(F(x,y,z)=0\), tangent at \((x_1,y_1,z_1)\): replace \(x^2\to xx_1\), \(y^2\to yy_1\), \(z^2\to zz_1\), \(xy\to\tfrac{1}{2}(xy_1+yx_1)\), etc.

e.g. Ellipsoid: tangent at \((x_1,y_1,z_1)\) is \(\dfrac{xx_1}{a^2}+\dfrac{yy_1}{b^2}+\dfrac{zz_1}{c^2}=1\).

Ruled Surfaces & Generators

Hyperboloid of one sheet and hyperbolic paraboloid are doubly ruled (two families of lines lying on them).

Generators of \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}-\dfrac{z^2}{c^2}=1\):

Family 1: \(\dfrac{x}{a}+\dfrac{z}{c}=\lambda\dfrac{1-y/b}{1}\), Family 2: similarly with \(\mu\).