The equation of the circle is
a) (x+ 1)² + ( y - 2)² = 36
or
x² + y² + 2x - 4y = 31
b) The points of intersection are ( -1.5 , - 4) and ( 3.5, 6)
STEP - BY - STEP EXPLANATION
What to find?
• Equation of a circle.
,
• Sketch of the circle representing the equation.
,
• Sketch of the line equation.
,
• Point of intersection between the circle and the line.
Given:
• Radius of the circle (r) =6
,
• Center of the circle(h, k) = (-1, 2)
,
• Equation of a straight line y=2x -1
To form the equation of the circle, we will first need to recall the standard circle equation below:
(x- h)² + (y-k)² = r²
Where
(h, k) is the center of the circle.
r is the radius of the circle.
From the information given, r = 6 h=-1 and k=2
Substitute the values into the formula.
(x+ 1)² + ( y - 2)² = 6²
Open the parenthesis.
x² + 2x + 1 + y² -4y + 4 = 36
Re-arrange and simplify.
x² + 2x + y² - 4y + 1 + 4 = 36
x² + y² + 2x - 4y + 5 = 36
x² + y² + 2x - 4y = 36 - 5
x² + y² + 2x - 4y = 31
Hence, the equation of the circle is
(x+ 1)² + ( y - 2)² = 36
or
x² + y² + 2x - 4y = 31
b) Find atleast two points on line y=2x -1
To do that, we need to find the x and y - intercept of the given equation.
x - intercept
Put y = 0 and solve for x
0 = 2x -1
2x = 1
x = 1/2
The x - intercept is (1/2 , 0).
y - intercept
Put x=0 and solve for y.
y = 2(0) - 1
y = 0 - 1
y = -1
The y - intercept is (0, -1).
The points on the line are (0.5, 0) and (0, -1)
We can now proceed to sketch the graph of both the circle and the straight line.
Note that:
To sketch the straight line, we will use the two points (0.5, 0) and (0, -1).
To sketch the circle, we will use: center (-1, 2) and radius = 6.
Attached below is the sketch of the graph.
Observe that point A and B are the points of intersection.
Therefore, the points of intersection are ( -1.5 , - 4) and ( 3.5, 6)