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the branch of Mathematics which deals with the properties, behaviors, and solutions of points, lines, curves, angles and surfaces and solids by means of algebraic methods in relation to a coordinate system. - Divided Divided into two two parts: (deals with figures on a plane surface) surf ace) and . Cartesian Coordinate system, Cartesian Coordinate Axes (x-axis and y-axis), Point 0 (origin), Quadrants, Abscissa or x-coordinate (distance from the the y-axis), Ordinate or y-coordinate (distance from the x-axis), rectangular coordinates or Cartesian coordinates or simply coordinates, plotting
positive negative
* distance can never be * When proving that the points are within the same line, it first to know the positioning of the points. * Check units for * A point that that is 3 units from the y - axis, . * A points that is 3 units from the x-axis * When asked to get the area of an irregular polygon, * The radius of a circle is 5 and its center is at (-3, -4). Find the length of the card that is bisected at (-5.5,-6.5).
* They are measured in the same way, hence, same signs. P1 --> P --> P2 * P1P and PP2 are measured oppositely, hence, opposite signs. P1 --> P2 --> P * two points that trisect the line *On the line joining (4, -5) to (-4, -2) , find the points which is three seventh the distance from the first to the second point. R1 = 3/7 and R2 = 4/7. *When ratio is given, it is the r1 /r2 itself.
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*Smallest positive angle that the straight line makes with the positive axis (α) *Measured from the positive axis in a counter-clockwise direction and is never greater than 180 degrees. *Can never be equal to 90 degrees d egrees
m = tan α *If between 0 and 90 degrees, *If between 90 and 180 degrees,
if lines are parallel *Three vertices of a parallelogram are x, y, z. Find the fourth vertex.
* right angle (perpendicular)
*if lines are perpendicular
Area of Triangle by Coordinates
line formed by connecting a vertex to another side bisecting it. = line formed by connecting a vertex to another side making a
1. Plot the points first. 2. Arrange the points counter clockwise based on the graphs. 3. Do the matrix calculation.
Line Parallel to an Axis
*Straight line is simply called line *x = x1 --> At a directed distance x1, if the line is parallel to the y-axis or perpendicular to the x-axis *y = y1 --> At a directed distance y1, if the line is parallel to the x-axis or perpendicular to the y-axis
General Equation Equation of a Line Ax + By + C = 0 * C has to be on the left side. * X has to be positive First Standard Equation of a Line = Point-Slope Form (PSF) y-y1 = m (x - x1) *m = slope *P (x1, y1) is any point in the line Second Standard Standard Equation of a Line = SlopeSlope- Intercept Form (SIF) y = mx + b *m = slope *b = y-intercept or (0, b) Equations of Parallel Lines *If two linear equations have identical x-coefficients and identical y-coefficient, the lines represented are parallel. Parallel lines: Ax + By + C 1 = 0 Ax + By + C 2 = 0 Equations of Perpendicular Lines
*If in two line equations, x-coefficient of the first is equal to the y-coefficient of the second and the ycoefficient of the first is numerically equal but of opposite sign to the x-coefficient of the second, or vice versa, the lines represented are perpendicular to each other. *Negative reciprocal. Ax + By + C 1 = 0 Bx - Ay + C2 = 0
Third Standard Equation of a Line = Intercept Form (IF)
*a = x-intercept or (x, 0) *b = y-intercept or (0,y)
Fourth Standard Equation of a Line = Normal Form (NF) xcosΘ + ysinΘ = P *Θ = angle of inclination *P = distance from the origin (0,0) Reduction of the General Form to the Normal Form
*where sign of radicand depends on the sign of B Distance from a line to a Point
*where sign of radicand depends on the sign of B Notes: *Bisector: use d1 = d2 and use P(x,y). *Ratio: d1 = rd2 *Product: d1d2=P
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Conic Sections: The section obtained when a plane is made to cut a right circular cone. cone . *Defined as the path of a point which moves so that its distance from a fixed point called the focus is in a constant ratio (eccentricity, e) to its distance from a fixed line called the directrix.
Equation of the Conic: FP = e*SP *Shape of the conic section depends on the position of the cutting plane 1. Circle - Parallel to the base. e --> 0 2. Parabola - parallel parallel to a plan tangent tangent to the cone. e = 1 3. Ellipse Ellipse - not parallel to a place tangent to the cone e < 1 4. Hyperbola - intersecting both nappes nappes (one of the two pieces of a double double cone) e > 1 5. Degenerate conics (point-ellipse, two coincident lines and tw o intersection lines) - passes through through the the vertex V.
* Note that e = 0, the definition fails. fails. Latus Rectum: the line through the focus that is parallel to the directrix intersecting the curve at R 1 and R2. Axis of the Conic: the line through F perpendicular to the directrix Vertex: point where axis of the conic intersects the conic itself. Parabola: conic section whose eccentricity is 1, locus of points which are equidistant from a fixed point and a fixed line. Equations of the Parabola: 4p = length of latus rectum 2p = distance between focus and R 1 and focus and R2 p = distance between vertex and focus, focus , and vertex and directrix Center Center at at (0,0) (0,0) Center Center at at (h,k) (h,k) Rightwards y2=4px
(y-k)2 = 4p(x-h)2
Leftwards
y2=-4px
(y-k)2 = -4p(x-h)2
Upwards
x2=4py
(x-h)2 = 4p(y-k)2
Downwards x2=-4py
(x-h)2 = 4p(y-k)2
Notes: 1. To know that you're correct, try substituting the values of R 1 and R2 on the equation since they are points in the conic. 2. If it says that the axis is parallel to the x-axis, x-axis , that means the parabola is either going leftwards or rightwards. rightwards . 3. If it says that the axis is parallel to the y-axis, that means the parabola is either going upwards or downwards. 4. Beware of getting the square roots. + or -. 5. Beware of the signs (especially if there are two answers). General Equation of the Parabola: *Parallel to the X-Axis: y2+Dy+Ex+F=0 *Parallel to the Y-Axis: x2+Dx+Ey+F=0
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- is the locus of a point which moves moves at a constant distance from a fixed point called its constant distance at any point along the circle from the center.
If right hand side of the equation
. If
there is
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A circle has its center on the line and tangent to the Find its equation. 2 2 x + y - 8x - 12y +16 +16 = 0 If its to x-axis at point (x,0), then the center is at ( x, y). Same x-value. If its to y-axis at point (0,y) then the center is a t (x,y). Same y-value. What is the equation of a circle passing through (12 ,1) and (2,-3) with center on the line 2x-5y+10=0. CP1 = CP2.
2x-5y+10=0
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Ellipse - a conic whose eccentricity is less than 1, that is, if P is any point on the ellipse,
Major Axis (2a) = Line segment V1V3, contains the two foci, always greater than the minor axis Minor Axis (2b) = Line segment V2V4 Properties of the Ellipse 1. The ellipse is a closed curve and symmetrical with respect to both its axes. axes. 2. The sum of the focal distances of any point on the ellipse is constant and equal to the length of the major axis: PF1 + PF2 = 2a. 2a. 3. As a corollary to the preceding property, we see that the distance fro m a focus to a vertex at one end of the minor axis is equal to half the length of the major axis: F2V2 = a 5. center to a directrix = a/e. 6. Distance of the center from foci (c or ae):
7. The length of a latera recta = 2b2 / a 8. e = c/a
Equations:
Properties of:
Problems: 1. Don't get confused with semi-major and major axes!
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Hyperbola - conic whose eccentricity is greater than 1 (e > 1)
Properties: 1. The hyperbola consists of two open branches , and is symmetrical with respect to both its axes . 2. The difference between the focal distances of any point on the hyperbola is constant and is equal to the length of the transverse axis: PF1 - PF2 = 2a 3. The distances from the center to a focus: 4. and center to a directrix:
5. q 6. The length of a latus rectum is 2b2/a 7. The diagonals (prolonged) of the rectangle of sides 2a and 2b and parallel to the transverse and conjugate axes respectively are asymptotes of the hyperbola. b2 = a2 (e2-1)
*Rectangular hyperbola if a = b. *Hyperbolas and ellipses are also called central conics because they possess centers while parabolas do not.
