Coordinate Geometry Transformations, Conics, and 3D Analytic Geometry Notes

Change of Origin (Translation of Axes):
- The origin is shifted from $(0,0)$ to a new point (\text{\alpha, \beta}) while keeping the axes parallel to the original axes.
- Relationship between old coordinates $(x, y)$ and new coordinates $(x', y')$ :
  - $x = x' + \alpha$
  - $y = y' + \beta$
  - $x' = x - \alpha$
  - $y' = y - \beta$
Rotation of Axes:
- The origin remains fixed, but the axes are rotated through an angle $\theta$ in the counter-clockwise direction.
- From geometric derivation (using triangles $OQN$ and $PQQ'$ ):
  - $x = x' \cos(\theta) - y' \sin(\theta)$
  - $y = x' \sin(\theta) + y' \cos(\theta)$
- Matrix Representation: $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix} \begin{pmatrix} x' \\ y' \end{pmatrix}$
  - The matrix $A = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}$ is an orthogonal matrix, satisfying $AA^T = I$ .
Origin and Axis Shift Combined:
- If the origin is shifted to $(\alpha, \beta)$ and the axes are rotated by value $\theta$ , the equations become:
  - $x = x' \cos(\theta) - y' \sin(\theta) + \alpha$
  - $y = x' \sin(\theta) + y' \cos(\theta) + \beta$

Removal of First-Degree Terms:
- To remove the terms involving $x$ and $y$ from an equation like $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ , the origin is shifted to the center of the conic $(\alpha, \beta)$ .
- Example Case: Remove first-degree terms in $3x^2 + 4y^2 - 12x + 4y + 13 = 0$ .
  - Substitute $x = x' + \alpha$ and $y = y' + \beta$ .
  - Expand: $3(x' + \alpha)^2 + 4(y' + \beta)^2 - 12(x' + \alpha) + 4(y' + \beta) + 13 = 0$ .
  - Collect coefficients of $x'$ and $y'$ :
    - Coeff of $x'$ : $6\alpha - 12 = 0 \implies \alpha = 2$
    - Coeff of $y'$ : $8\beta + 4 = 0 \implies \beta = -\frac{1}{2}$
  - New equation: $3(x')^2 + 4(y')^2 + (3(2)^2 + 4(-\frac{1}{2})^2 - 12(2) + 4(-\frac{1}{2}) + 13) = 0$
  - Final result: $3(x')^2 + 4(y')^2= 0$ .
Removal of the $xy$ Term:
- To remove the product term $xy$ from the general equation $ax^2 + 2hxy + by^2$ , the axes are rotated by an angle $\theta$ .
- Substitution: $x = x_1 \cos(\theta) - y_1 \sin(\theta)$ and $y = x_1 \sin(\theta) + y_1 \cos(\theta)$ .
- The angle $\theta$ is chosen such that the coefficient of $x_1 y_1$ is zero:
  - Coefficient: $(b - a) \sin(2\theta) + 2h \cos(2\theta) = 0$
  - $\tan(2\theta) = \frac{2h}{a - b}$
  - $\theta = \frac{1}{2} \tan^{-1}\left(\frac{2h}{a - b}\right)$

The Invariants:
- When axes are rotated, certain combinations of coefficients remain constant:
  - $a + b = A + B$
  - $ab - h^2 = AB - H^2$
Condition for a Pair of Straight Lines:
- The general equation $ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0$ represents a pair of straight lines if the discriminant $\Delta = 0$ : $\begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} = 0$
  - Expanded as: $abc + 2fgh - af^2 - bg^2 - ch^2 = 0$ .
Center of a Conic:
- Determined by partial derivatives: $\frac{\partial f}{\partial x} = 0$ and $\frac{\partial f}{\partial y} = 0$ .
- $ax + hy + g = 0$
- $hx + by + f = 0$
- Solving these gives the center $(\alpha, \beta)$ :
  - $\alpha = \frac{hf - bg}{ab - h^2}$
  - $\beta = \frac{gh - af}{ab - h^2}$
Standard Form Reduction Steps:
1. Find the center $(\alpha, \beta)$ and shift the origin.
2. Calculate the new constant $C_1 = g\alpha + f\beta + c$ .
3. Rotate axes to eliminate the $xy$ term using survivors $a_1, b_1$ where $a_1 + b_1 = a + b$ and $a_1 b_1 = ab - h^2$ .
Classification by Discriminant ( $ab - h^2$ ):
- If ab - h^2 > 0: The conic is an Ellipse.
- If ab - h^2 < 0: The conic is a Hyperbola.
- If $ab - h^2 = 0$ : The conic is a Parabola.

Direction Cosines (DC):
- If $\alpha, \beta, \gamma$ are angles made by a line with positive $x, y, z$ axes:
  - $l = \cos(\alpha)$ , $m = \cos(\beta)$ , $n = \cos(\gamma)$
- Property: $l^2 + m^2 + n^2 = 1$ .
- For a point $P(x, y, z)$ with distance $r$ from the origin:
  - $x = rl$ , $y = rm$ , $z = rn$
Direction Ratios (DR):
- Any three numbers $a, b, c$ proportional to $l, m, n$ .
- Relationship: $l = \frac{\pm a}{\sqrt{a^2 + b^2 + c^2}}$ , $m = \frac{\pm b}{\sqrt{a^2 + b^2 + c^2}}$ , $n = \frac{\pm c}{\sqrt{a^2 + b^2 + c^2}}$ .
Angle Between Two Lines:
- If lines have DCs $(l_1, m_1, n_1)$ and $(l_2, m_2, n_2)$ , the angle $\theta$ is:
  - $\cos(\theta) = l_1 l_2 + m_1 m_2 + n_1 n_2$
- Perpendicular Lines: $l_1 l_2 + m_1 m_2 + n_1 n_2 = 0$ .
- Parallel Lines: $\frac{l_1}{l_2} = \frac{m_1}{m_2} = \frac{n_1}{n_2}$ .

Equation of a Plane:
- General Form: $ax + by + cz + d = 0$
- Passing through point $(x_1, y_1, z_1)$ : $a(x - x_1) + b(y - y_1) + c(z - z_1) = 0$
- Intercept Form: $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ .
- Normal Form: $lx + my + nz = p$ .
Angle between a Line and a Plane:
- If line has DRs $(l, m, n)$ and plane has normal DRs $(a, b, c)$ , the angle $\theta$ is given by:
  - $\sin(\theta) = \frac{al + bm + cn}{\sqrt{a^2 + b^2 + c^2}\sqrt{l^2 + m^2 + n^2}}$
Shortest Distance (SD) between two lines:
- Calculated as the projection of the line segment joining points on the two lines onto the common perpendicular.

Equation of a Sphere:
- General Form: $x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0$
- Center: $(-u, -v, -w)$ .
- Radius: $r = \sqrt{u^2 + v^2 + w^2 - d}$ .
Orthogonal Spheres:
- Two spheres are orthogonal if they intersect such that their tangent planes at the point of intersection are perpendicular.
- Condition: $2u_1 u_2 + 2v_1 v_2 + 2w_1 w_2 = d_1 + d_2$ .