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CHAPTER 21 KEY FORMULAS AND EQUATIONS

Distance formula	Fig. 21.2	$d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$ $d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$ (21.1)
Slope	Fig. 21.4	$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ (21.2)
	Fig. 21.7	$m = \tan α (0 ° \leq α < 180 °)$ $m = \tan α (0 ° \leq α < 180 °)$ (21.3)
	Fig. 21.9	$m_{1} = m_{2} (for \| \| lines)$ $m_{1} = m_{2} (for \| \| lines)$ (21.4)
	Fig. 21.10	$m_{2} = - \frac{1}{m_{1}} or m_{1} m_{2} = - 1 (for ⊥ lines)$ $m_{2} = - \frac{1}{m_{1}} or m_{1} m_{2} = - 1 (for ⊥ lines)$ (21.5)
Straight line	Fig. 21.15	$y - y_{1} = m (x - x_{1})$ $y - y_{1} = m (x - x_{1})$ (21.6)
	Fig. 21.18	$x = a$ $x = a$ (21.7)
	Fig. 21.19	$y = b$ $y = b$ (21.8)
	Fig. 21.22	$y = m x + b$ $y = m x + b$ (21.9)
		$A x + B y + C = 0$ $A x + B y + C = 0$ (21.10)
Circle	Fig. 21.30	${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ (21.11)
	Fig. 21.33	$x^{2} + y^{2} = r^{2}$ $x^{2} + y^{2} = r^{2}$ (21.12)
		$x^{2} + y^{2} + D x + E y + F = 0$ $x^{2} + y^{2} + D x + E y + F = 0$ (21.14)
Parabola	Fig. 21.43	$y^{2} = 4 p x$ $y^{2} = 4 p x$ (21.15)
	Fig. 21.46	$x^{2} = 4 p y$ $x^{2} = 4 p y$ (21.16)
Ellipse	Fig. 21.58	$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ (21.17)
	Fig. 21.58	$a^{2} = b^{2} + c^{2}$ $a^{2} = b^{2} + c^{2}$ (21.18)
	Fig. 21.59	$\frac{y^{2}}{a^{2}} + \frac{x^{2}}{b^{2}} = 1$ $\frac{y^{2}}{a^{2}} + \frac{x^{2}}{b^{2}} = 1$ (21.19)
Hyperbola	Fig. 21.71	$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ (21.20)
	Fig. 21.71	$c^{2} = a^{2} + b^{2}$ $c^{2} = a^{2} + b^{2}$ (21.21)
	Fig. 21.70	$y = \pm \frac{b x}{a} (asymptotes)$ $y = \pm \frac{b x}{a} (asymptotes)$ (21.23)
	Fig. 21.72	$\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1$ $\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1$ (21.24)
	Fig. 21.78	$x y = c$ $x y = c$ (21.25)
Translation of axes	Fig. 21.83	$x = x^{'} + h and y = y^{'} + k$ $x = x^{'} + h and y = y^{'} + k$ (21.26)
		$x^{'} = x - h and y^{'} = y - k$ $x^{'} = x - h and y^{'} = y - k$ (21.27)
Parabola, vertex (*h, k*)		${(y - k)}^{2} = 4 p (x - h) (axis parallel to x -axis)$ ${(y - k)}^{2} = 4 p (x - h) (axis parallel to x -axis)$ (21.28)
		${(x - h)}^{2} = 4 p (y - k) (axis parallel to y - axis)$ ${(x - h)}^{2} = 4 p (y - k) (axis parallel to y - axis)$ (21.29)
Ellipse, center (*h, k*)		$\frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - k)}^{2}}{b^{2}} = 1 (major axis parallel to x -axis)$ $\frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - k)}^{2}}{b^{2}} = 1 (major axis parallel to x -axis)$ (21.30)
		$\frac{{(y - k)}^{2}}{a^{2}} + \frac{{(x - h)}^{2}}{b^{2}} = 1 (major axis parallel to y -axis)$ $\frac{{(y - k)}^{2}}{a^{2}} + \frac{{(x - h)}^{2}}{b^{2}} = 1 (major axis parallel to y -axis)$ (21.31)
Hyperbola, center (*h, k*)		$\frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1 (transverse axis parallel to x -axis)$ $\frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1 (transverse axis parallel to x -axis)$ (21.32)
		$\frac{{(y - k)}^{2}}{a^{2}} - \frac{{(x - h)}^{2}}{b^{2}} = 1 (transverse axis parallel to y -axis)$ $\frac{{(y - k)}^{2}}{a^{2}} - \frac{{(x - h)}^{2}}{b^{2}} = 1 (transverse axis parallel to y -axis)$ (21.33)
Second-degree equation		$A x^{2} + B x y + C y^{2} + D x + E y + F = 0$ $A x^{2} + B x y + C y^{2} + D x + E y + F = 0$ (21.34)
Rotation of axes	Fig. 21.97	$\begin{array}{rcl} x & = & x^{'} \cos θ - y^{'} \sin θ \\ y & = & x^{'} \sin θ + y^{'} \cos θ \end{array}$ $\begin{array}{rcl} x & = & x^{'} \cos θ - y^{'} \sin θ \\ y & = & x^{'} \sin θ + y^{'} \cos θ \end{array}$
Angle of rotation		$\tan 2 θ = \frac{B}{A - C} (A \neq C)$ $\tan 2 θ = \frac{B}{A - C} (A \neq C)$ (21.39)
		$θ = 45 ° (A = C)$ $θ = 45 ° (A = C)$ (21.40)
Polar coordinates	Fig. 21.106	$x = r \cos θ y = r \sin θ$ $x = r \cos θ y = r \sin θ$ (21.41)
		$\tan θ = \frac{y}{x} r = \sqrt{x^{2} + y^{2}}$ $\tan θ = \frac{y}{x} r = \sqrt{x^{2} + y^{2}}$ (21.42)

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Table of Contents for CHAPTER 21 KEY FORMULAS AND EQUATIONS

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Table of Contents for
CHAPTER 21 KEY FORMULAS AND EQUATIONS