Coordinate Geometry Practice I

length of line AB

√[(x₂-x₁)² + (y₂-y₁)²]

midpoint of line AB

([(x₁+x₂)/2], [(y₁+y₂)/2])

gradient of line AB

m = (y₂-y₁)/(x₂-x₁)

equation of a line passing through A & B

y-y₁ = m(x-x₁)

product of the gradients of perpendicular lines

-1

equation of a circle with a centre (a, b) and radius, r

(x-a)² + (y-b)² = r² → x² + y² - 2ax - 2bx + c = 0

formula for the centre of the circle

(-1/2 coefficient of x, -1/2 coefficeint of y)

formula for c

a² + b² - r²

State the steps in determining whether or not a line intersects with a circle

1. Substitute the value of y from the equation of the line into the circle’s equation

2. After expanding, use the values of a, b and c to get the equation b² - 4ac

3. If b²-4ac is less than 0, it does not intersect with the circle
   If b²-4ac is equal to 0, it is a tangent to the circle
   If b²-4ac is greater than 0, it intersects with the circle at 2 points

State the steps in determining the points of intersection between two circles

1. Make a simultaneous equation with the equations of the two circles and get a value for either x or y

2. Substitute the value obtained into the equation of any circle and solve the created quadratic equation

3. Use the solution of the quadratic equation to get the coordinates of the points of intersection using the equation from step 1

parabola

set of all points in a plane which are the same distance from a fixed line

equation of parabola parallel to the x-axis

x = ay² + by + c, a ≠ 0

equation of parabola parallel to the y-axis

y = ax² + bx + c, a ≠ 0

ellipse

locus of points in which the sum of the distances from any two fixed pointsis constant

equation of an ellipse with centre (0, 0) foci at (-c, 0) and (c, 0), vertices at (-a, 0) and (a, 0) and the major axis in the x-axis

(x²/a²) + (y²/b²) = 1

equation of an ellipse with centre (0, 0) foci at (0, -c) and (0, c), vertices at (0, -a) and (0, a) and the major axis in the y-axis

(x²/b²) + (y²/a²) = 1

formula for c in an ellipse

c² = a² - b²

State the steps for determining the point of intersection between a curve and a line

1. Equate the two equations

2. Find a value for x/y

3. Substitute the obtain value(s) into any of the equations to get the coordinates of the point(s) of intersection

State the steps for determining the point(s) of intersection between two curves

1. Use one of the equations to find a value for x/y

2. Substitute that value into any one of the equations to get the counterpart values

3. Substitute the obtained value(s) into any of the equations to get the coordinates of the point(s) of intersection

parametric equation of a curve

values of the functions y = g(t) and x = f(t) when t is a parameter

Cartesian equation of a curve

combination of two parametric equations so that y = f(x)

How do you find the Cartesian equation of a curve?

1. Obtain a value for t using one of the parametric equation

2. Substitute obtained value into the other parametric equation

parametric equations of a circle

x = a + rcos(t) and y = b + rcos(t)

Cartesian equation of a circle with centre (a, b)

(x-a)² + (y-b)² = r²

How do you find the Cartesian equation of a curve?

1. Get the parametric equations in terms of their lone trigonometric functions

2. Add the equations so that they add up to 1

3. Simplify any fractions

parametric equations of an ellipse with centre (h, k)

x = h + acos(t) and y = k + bsin(t)

Cartesian equation of an ellipse with centre (h, k)

[(x-h)²/a²] + [(y-k)²/b²] = 1

How do you find the Cartesian equation of an ellipse?

1. Get the parametric equations in terms of their lone trigonometric functions

2. Add the equations so that they add up to 1

parametric equation of a parabola with centre (h, k) opening to the right

y = k + pt, x = h + pt², where p is the distance from the vertex to the focus

parametric equation of a parabola with centre (h, k) opening to the left

y = k + pt, x = h - pt², where p is the distance from the vertex to the focus

Cartesian equation of a parabola opening to the right with centre (h, k)

(y-k)² = 4px, where p is the distance from the vertex to the focus

Cartesian equation of a parabola opening to the left with centre (h, k)

(y-k)² = -4p(x-h), where p is the distance from the vertex to the focus

How to determine loci satisfying points?

If points are equidistant to a line and a point → parabola

If a constant sum between points is involved → ellipse

If there is a fixed point → circle