Analytic Geometry Flashcards Preview

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Flashcards in Analytic Geometry Deck (24):

sides of a 45, 45, 90 triangle

x, y, (√2 *x)


sides of a 30, 60, 90

short side= x

longer side = √3*x

hyootenuse = 2x


what is a real number?

A real number is a number whose square is a positive number.


standard equation form of a hyperbola opening right/left

((x-h)2/a2) - ((y-k)2/b2) = 1

center of hyperbola (asymptote intersection) at (h,k)


standard equation form of a hyperbola opening up/down


what is the asymptote slope

((y-k)2/b2) -  ((x-h)2/a2) = 1

center of hyperbola (asymptote intersection) at (h,k)


a is distance from center to vertex

asymptote slope = b/a


Given a hyperbola, if the negative sign is in front of the x variable, it opens ____/______, if the negative sign is in front of the y variable, it opens ____/______

 x= up/down

 y = right/left


hyperbola asymptote equation when opens

- left/right

- up.down

 left/right   y = k±(b/a)(x-h) 

up/down    y = k±(b/a)(x-h)


they are the same


what is a conic section?

a curve obtained as the intersection of a cone (more precisely, a right circular conical surface) with a plane


general formula for an ellipse

((x-h)2/a2) + ((y-k)2/b2) = 1


Given eelipse std equation, the coordinates for the left-most point, the right-most point, upper most point, and lower most point

right most= (h+a, k)

left most= (h-a,k)

upper most= (h, k+b)

lower most= (h, k-b)


parabolic functions,

standard form, others

f(x) = ax2 + bx +c

f(x) = a(x-h)2+k

(h,k) is vertex, ±a is open up/down


how to find h,k (vertex) with parabola standard form equation

h = ((-b/2a), (f(-b/2a))


when completng the square, what do you do to the "b" term to get the new (temp) c term that will allow you to complete the square

divide by 2 then square


formula for the distance between two points in cartesian plane and circle standard form equation

the distance formula

d = ((x1-x)2 + (y1-y)2)(1/2)

for a circle:

r = ((x1-h)2 + (y1-k)2)(1/2) 


r2 = (x-h)2 + (y-k)2

where (h,k) is the center


If trying to find the equation for a circle, but given this, what are the steps to get this into standard form

x2 + y2 + 8x + 7

get constant on other side

group the x and y terms together

complete square for relevant variable(s)


linear equations/lines adhere to 4 rules

1) variables x and y are raised only to the 1st power

2) variables may be multiplied only by real numbers

3) any real # term may be added or subtracted

4) Nothing else is permitted (1/x, ex, x2)


what is needed to define a line equation?

1) 2 points or;

2) one point and a slope


3 different ways an equation may be written for a line

1) point intercept y=mx+b

2) point slope form y-y= m(x-x1)

3) two-point form  y-y1 = ((y2-y1)/(x2-x1))(x-x1)


4 methods to solve quadratic equation

1) factoring

2) quadratic equation

3) completing the square

4) factor by grouping


what is the discriminant in a quadratic equation

b2 - 4ac


what the discriminant tells you about the solution to a quardratic equation

- if <0, then there are 2 distinct real solutions

- if = 0, then one real # solution

- if > 0, then 2 distinct complex imaginary number solutions



when is a something a function

for all values of x there is only one value for y


law of sines

Sin A/a = Sin B/b = Sin C/c


when to use the law of consines (2)

1) When the 3rd side of a triangle is needed, you know the other two sides, and the angle between

2) When the angles of a triangle are needed and you know all three sides