Goemetry Of Lines And Circles Flashcards
What are the three equations of a line
y-y1 = m(x-x1)
y= mx + c
ax+by+c =0 (abc are integers)
Easiest way to find y-y1 = m(x-x1)
Label the points (x1,y1) and (x2,y2)
Gradient find it and call it m. M=y2-y1/x2-x1
Write the equation using y-y1 = m(x-x1)
Convert to one of the other forms of necessary.
Finding a gradient
M=y2-y1/x2-x1
Rearranging y-y1 = m(x-x1)
For the form y=mx+c take everything except y to one side
To find the form ax+by+c=0 take everything over to one side. And then get rid of the fractions
If you end up with two fractions with different denominators. Times the fraction by their LCM
Line segment
And finding the mid point
Is just a straight line that goes between two points.
Midpoint AB = (xA+xB/2,yA+yB/2)
Finding the length of a line segment
If question says exact leave in surd form
Use pythagoras’ theorem
Length= distance between two points.
Length AB = #/(xB-xA)^2+(yB-yA)^2
Parallel lines have equal gradients
Question that ask you to find parallel lines are easier with the ax+by+c=0 equation.
Just replace the y intercept with k or c.
Then substitute values given in question into equation to find y intercept of parallel line.
The gradient of a perpendicular line or normal is
-1 % the gradient of the other line
If you start with the line equation ax+by+c=0 swap the coefficients of the x and y value around and change the sign of one of them.
Equation of a circle
(X-a)^2+(y-b)^2=r^2
Based on Pythagoras.
Complete the square to get into the familiar form of the circle equation.
Not all circle equations look like (X-a)^2+(y-b)^2=r^2
Collect the x and y terms together.
Complete the squares. Until equation looks like the one above
Properties of circles
Draw them to remember better.
The angle in a semicircle is a right angle.
The perpendicular from the centre to a chord bisects the chord.
A radius and a tangent to the same point will meet at right angles.
Use the gradient rule for perpendicular lines
Sketch with questions of the sort.
M=y2-y1/x2-x1
