Analytic Geometry - 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 into two parts: Plane Analytic Geometry (deals with figures on a plane surface) and Solid Analytic Geometry.- Terms: Cartesian Coordinate system, Cartesian Coordinate Axes (x-axis and y-axis), Point 0 (origin), Quadrants, Abscissa or x-coordinate (distance from the y-axis), Ordinate or y-coordinate (distance from the x-axis), rectangular coordinates or Cartesian coordinates or simply coordinates, plotting

Directed Line Segments*Left to right: positive*Right to left: negative

Distance Between Two Points:

* distance can never be negative* When proving that the points are within the same line, plot it first to know the positioning of the points.* Check units for area = square units* A point that is 3 units from the y - axis, (3, y).* A points that is 3 units from the x-axis (x, 3)* When asked to get the area of an irregular polygon, divide them into parts!* 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). Use radius as the hypotenuse and the distance from center to point as one of the legs. The answer is the other leg times 2.

Division of a Line Segment

*Internal: They are measured in the same way, hence, same signs. P1 --> P --> P2

*External: P1P and PP2 are measured oppositely, hence, opposite signs. P1 --> P2 --> P*Trisection points: 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.

Alternate Formula for Division of a Line Segment

Midpoint of a Line Segment

Analytic Geometry (Prelims)Tuesday, July 15, 2014 6:02 PM

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Midpoint of a Line Segment

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Angle of Inclination:*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

Slope of the Linem = tan α*If between 0 and 90 degrees, m (slope) is positive*If between 90 and 180 degrees, m (slope) is negative

Slope in terms of Coordinates

Slope of Parallel Linesm1=m2 if lines are parallel*Three vertices of a parallelogram are x, y, z. Find the fourth vertex. Get the slope of the other two pairs, get the equation of the third and fourth side and get the intersection.*Median (opposite side bisector) = line formed by connecting a vertex to another side bisecting it.*Altitudes (opposite side perpendicular) = line formed by connecting a vertex to another side making a right angle (perpendicular)

Slope of Perpendicular Lines

*if lines are perpendicular

Angle Formed by Two Lines

Plot the points first.1.Area of Triangle by Coordinates

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Plot the points first.1.Arrange the points counter clockwise based on the graphs.2.Do the matrix calculation.3.

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 of a LineAx + 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 Equation of a Line = Slope- 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 + C1 = 0Ax + By + C2 = 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 y-

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*If in two line equations, x-coefficient of the first is equal to the y-coefficient of the second and the y-coefficient 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 + C1 = 0Bx - 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 BNotes:*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. *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

Circle - Parallel to the base. e --> 01.Parabola - parallel to a plan tangent to the cone. e = 12.Ellipse - not parallel to a place tangent to the cone e < 13.Hyperbola - intersecting both nappes (one of the two pieces of a double cone) e > 14.Degenerate conics (point-ellipse, two coincident lines and two intersection lines) - passes through the vertex V.5.

*Shape of the conic section depends on the position of the cutting plane

* Note that e = 0, the definition fails.

Latus Rectum: the line through the focus that is parallel to the directrix intersecting the curve at R1 and R2.Axis of the Conic: the line through F perpendicular to the directrixVertex: 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 rectum2p = distance between focus and R1 and focus and R2

p = distance between vertex and focus, and vertex and directrix

Center at (0,0) Center at (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

To know that you're correct, try substituting the values of R1 and R2 on the equation since they are points in the conic.1.If it says that the axis is parallel to the x-axis, that means the parabola is either going leftwards or rightwards.2.If it says that the axis is parallel to the y-axis, that means the parabola is either going upwards or downwards.3.Beware of getting the square roots. + or -. 4.Beware of the signs (especially if there are two answers).5.

Notes:

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

Chapter 6 - Conic SectionsMonday, August 25, 2014 12:19 PM

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Circle - is the locus of a point which moves at a constant distance from a fixed point called its center.Radius - constant distance at any point along the circle from the center.

Standard Equation:

*If right hand side of the equation (r2) is zero, locus is a point. If r2 is negative, there is no locus.

General Equation:

Answer: x2 + y2 - 8x - 12y +16 = 0a.If its tangent to x-axis at point (x,0), then the center is at (x, y). Same x-value.b.If its tangent to y-axis at point (0,y) then the center is at (x,y). Same y-value.c.

A circle has its center on the line 2y = 3x and tangent to the x-axis at (4,0). Find its equation.1.

Answer: c(5,4)a.First equation: CP1 = CP2. Second equation: 2x-5y+10=0b.

What is the equation of a circle passing through (12,1) and (2,-3) with center on the line 2x-5y+10=0.2.

Problems:

CircleMonday, August 04, 2014 8:35 PM

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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 axisMinor Axis (2b) = Line segment V2V4

The ellipse is a closed curve and symmetrical with respect to both its axes.1.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.2.As a corollary to the preceding property, we see that the distance from a focus to a vertex at one end of the minor axis is equal to half the length of the major axis: F2V2 = a

3.

center to a directrix = a/e.5.Distance of the center from foci (c or ae): 6.

Properties of the Ellipse

The length of a latera recta = 2b2 / a7.e = c/a8.

Equations:

Properties of:

Don't get confused with semi-major and major axes! 1.Problems:

The EllipseThursday, August 21, 2014 12:44 PM

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Hyperbola - conic whose eccentricity is greater than 1 (e > 1)

The hyperbola consists of two open branches, and is symmetrical with respect to both its axes.1.

PF1 - PF2 = 2aThe difference between the focal distances of any point on the hyperbola is constant and is equal to the length of the transverse axis:2.

The distances from the center to a focus:3.

and center to a directrix:4.

q5.The length of a latus rectum is 2b2/a6.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.

7.

Properties:

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.

HyperbolaWednesday, September 24, 2014 8:11 PM

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