Photo AI
The circle C has equation $x^2 + y^2 + 4x - 2y - 11 = 0$ Find (a) the coordinates of the centre of C, (b) the radius of C, (c) the coordinates of the points where C crosses the y-axis, giving your answers as simplified surds. - Edexcel - A-Level Maths Pure - Question 6 - 2011 - Paper 2
Question 6
The circle C has equation $x^2 + y^2 + 4x - 2y - 11 = 0$ Find (a) the coordinates of the centre of C, (b) the radius of C, (c) the coordinates of the points ... show full transcript
Worked Solution & Example Answer:The circle C has equation $x^2 + y^2 + 4x - 2y - 11 = 0$ Find (a) the coordinates of the centre of C, (b) the radius of C, (c) the coordinates of the points where C crosses the y-axis, giving your answers as simplified surds. - Edexcel - A-Level Maths Pure - Question 6 - 2011 - Paper 2
Step 1
the coordinates of the centre of C
Answer
To find the coordinates of the center of the circle, we first rewrite the equation in the standard form. Starting from:
we group the x and y terms:
Next, we complete the square for the x terms and the y terms. For the x terms:
For the y terms:
Putting these back into the equation gives us:
\Rightarrow (x + 2)^2 + (y - 1)^2 = 16 $$ From this standard form, the center of the circle C is at the coordinates (-2, 1).Step 2
Step 3
the coordinates of the points where C crosses the y-axis
Answer
To find where the circle crosses the y-axis, we set x = 0 in the equation.
Substituting x = 0 into the rearranged circle equation:
\Rightarrow 4 + (y - 1)^2 = 16 \ \Rightarrow (y - 1)^2 = 12 \ \Rightarrow y - 1 = ext{±} ext{sqrt}(12) \ \Rightarrow y = 1 ext{±} 2 ext{sqrt}(3) $$ Thus, the coordinates where the circle crosses the y-axis are: $$ (0, 1 + 2 ext{sqrt}(3)) \ (0, 1 - 2 ext{sqrt}(3)) $$