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Here we will be discussing the equations of circles along with the intersection of lines with circles.
2.
Higher Maths 2 4 2
Distance Between Two B ( x2 ,
y2 )
d = √( x 2– + ( y2 –
y2 – y1
x1)²The Distance
y1)²
Exampl Formula
Calculate the distance A x2 –
between (-2,9) and (4,- ( x1 ,
y1 ) x1
d = √ 6² + 12²
Where required, write
answers as a surd in its
= √ 18 = 6√ 5
simplest form.
0
3.
Higher Maths 2 4 3
Points on a
Plot the following points
and find a rule connecting x
and y.
(5,0) (4,3) (3,4) (0,5)
(-3 , 4 ) (-4 , 3 ) (-5 , 0 ) (-4 ,-3 )
(-3 ,-4 ) ( 0 ,-5 ) ( 3 ,-4 ) ( 4 ,-3 )
All points lie on a circle with radius
5 units and centre at the origin. For any
For any point on the x² + y² = x ²radius...
+ y² =
circle, 25 r²
4.
Higher Maths 2 4 4
The Equation of a Circle with centre at the
For any circle with
radius r and centre the The
origin, ‘Origin’
x² + y² = r is the origi
n
point
² (0,0)
Substitute point into
equation:
Show that the point (-73 , x ² + y= (-3)² + ( 7 )²
) ² = 9 + 7
lies on the circle with
x ² + y ² = 16 = 16
The point lies on the
circle.
5.
Higher Maths 2 4 5
The Equation of a Circle with centre
(a, b) r
Not all circles are centered at the
For any circle with radius (a,b)
r and centre at the point
( a , b ) ...
( x – a )² + ( y – b )² =
r²
Exampl ( x – a )² + ( y – b )² =
( xr–²3 ) ² + ( y – (-5) ) ² =2 (3
Write the equation of
)²
circle with centre ( 3 ,- ( x – 3 )² + ( y + 5 )² =
5) 12
6.
Higher Maths 2 4 6
The General Equation of a
Try expanding the equation of a circle with centre ( -
g, - f ).
( x + g )2 + ( y + f )2 =
r22 this is just
( x + 2g x + g ) + ( y + 2fy + f )
2 2 2
a
= r2 number...
c = g + f 2 – r2
2
x 2 + y 2 + 2g x + 2f y + g 2 + f 2 – r 2
= 0 r2 = g2 + f 2 – c
x 2 + y 2 + 2g x + 2f y + c
r= g2 + f 2 – c
= 0
General Equation of a Circle with
) radiusr =
center ( -g , - f and g2 + f 2 – c
7.
Higher Maths 2 4 7
Circles and Straight
A line and a circle can have two, one or no points of
two points one point no points
of of of
intersection intersection intersection
r
A line which intersects a circle at only
one point is at 90° to the radius and is is
called a tangent.
8.
Higher Maths 2 4 8
Intersection of a Line and a
How to find the points of intersection between a line
and a circle:
• rearrange the equation of the line into the form y=
mx + c
• substitute y = m x + c into the equation of the circle
Exampl y = 2x
• solve the quadratic for x and substitute x = m
into 3x+ orc -3
to y
the
find x 2 + (2 x)2 = 45 Substitute into y =
intersection of 2 x :y = 6
x 2 + 4 x 2 = 45 or -6
x 2 +circle
y2 = 45
5 x 2 = 45 Points of
2x – y = 0
x2 = 9 intersection are
and the line
(3,6) and (-3,-6).
9.
Higher Maths 2 4 9
Circles
Intersection of a Line and a Circle
(continued)
Example
2
Find where the line 2 x – y + 8 = 0
intersects the circle x2 + y2 + 4x + 2y
– 20 = 0
x 2 + (2 x + 8)2 + 4 x + 2 (2 x + 8) – 20 = 0
x2 + 4x2 + 32 x + 64 + 4 x + 4 x + 16 – 20 = 0
5 x 2 + 40 x + 60 = 0 x = -2 or -6
Factorise
5( x 2 + 8 x + 12 ) = 0 and Substituting into y=2
solve
x + 8 points of
5( x + 2)( x + 6) = 0 intersection as
(-2,4) and (-6,-4).
10.
Higher Maths 2 4 1
Circles 0
The Discriminant and
The discriminant can be x = -b ± b 2 – (4
used to show that a line is a
2a
ac )
y=
• substitute mx + c into the circle
b 2 – (4 ac )
• rearrange to form a quadratic equation Discrimina
nt
• 2evaluate the discriminant
b – (4 ac ) > 0 Two points of
intersection
b 2 – (4 ac ) = 0 The line is a tangent r
b 2 – (4 ac ) < 0 No points of
intersection
11.
Higher Maths 2 4 1
Circles 1
Circles and
Tangents
Example
Show that the line 3 x + y = -10 is a
tangent to the circle x2 + y2 – 8x + 4y
– 20 = 0
x + (-3 x – 10)2 – 8 x + 4 (-3 x – 10) – 20 = 0
2
x2 + 9x2 + 60 x + 100 – 8 x – 12 x – 40 – 20 = 0
10 x 2 + 40 x + 40 = 0
b 2 – (4 ac=) 40 2 – ( 4 × 10 × 40 ) The line is a
tangent to the
= 1600 – 1600
circle since
= 0 b2
– (4 ac ) = 0
12.
Higher Maths 2 4 1
Circles 2
Equation of
Tangents y – b = m( x – a)
To find the equation of
a tangent to a circle: Straight Line
Equation
• Find the center of the circle
and the y2 – y1
point where the tangent
m radius x2 – x1
• Calculate the gradient of the
intersects =
radius using the gradient formula
• Write down the gradient of the r
• Substitute the gradient of the
tangent –1
and the point of intersection m tangent =
m
y – b = m( x – a)
radius
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