A-Level Edexcel Maths Pure Solving Equations: The circle C has equation $x^{2}

The circle C has equation $x^{2} + y^{2} - 10x + 6y + 30 = 0$ Find (a) the coordinates of the centre of C, (b) the radius of C, (c) the y coordinates of the points where the circle C crosses the line with equation $x = 4$, giving your answers as simplified surds. - Edexcel - A-Level Maths Pure - Question 6 - 2017 - Paper 3

Question 6

$The-circle-C-has-equation-$x^{2}-+-y^{2}---10x-+-6y-+-30-=-0$--Find--(a)-the-coordinates-of-the-centre-of-C,--(b)-the-radius-of-C,--(c)-the-y-coordinates-of-the-points-where-the-circle-C-crosses-the-line-with-equation-$x-=-4$,-giving-your-answers-as-simplified-surds.-Edexcel-A-Level Maths Pure-Question 6-2017-Paper 3.png$

The circle C has equation $x^{2} + y^{2} - 10x + 6y + 30 = 0$ Find (a) the coordinates of the centre of C, (b) the radius of C, (c) the y coordinates of the poin... show full transcript

Worked Solution & Example Answer:The circle C has equation $x^{2} + y^{2} - 10x + 6y + 30 = 0$ Find (a) the coordinates of the centre of C, (b) the radius of C, (c) the y coordinates of the points where the circle C crosses the line with equation $x = 4$, giving your answers as simplified surds. - Edexcel - A-Level Maths Pure - Question 6 - 2017 - Paper 3

Step 1

the coordinates of the centre of C

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Answer

To find the coordinates of the centre of the circle C, we need to rewrite the equation in standard form. The given equation is:

$x^{2} + y^{2} - 10x + 6y + 30 = 0$

We can rearrange it by completing the square.
For the x-terms:

Start with $x^{2} - 10x$ .
Complete the square: $x^{2} - 10x = (x - 5)^{2} - 25$ .
For the y-terms:
Start with $y^{2} + 6y$ .
Complete the square: $y^{2} + 6y = (y + 3)^{2} - 9$ .

Substituting these back into the equation gives:

$(x - 5)^{2} - 25 + (y + 3)^{2} - 9 + 30 = 0$
Simplifying further, we find:
$(x - 5)^{2} + (y + 3)^{2} - 4 = 0$
Thus:
$(x - 5)^{2} + (y + 3)^{2} = 4$
This indicates that the centre of the circle is at the point $(5, -3)$ .

Step 2

the radius of C

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Answer

The radius can be determined from the standard form of the circle's equation. From
$(x - 5)^{2} + (y + 3)^{2} = 4$
we see that the radius $r$ is given by:
$r = ext{sqrt}(4) = 2.$
Thus, the radius of circle C is 2.

Step 3

the y coordinates of the points where the circle C crosses the line $x = 4$

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Answer

To find the y-coordinates where the circle crosses the line $x = 4$ , we substitute $x = 4$ into the circle's equation:
$(4 - 5)^{2} + (y + 3)^{2} = 4$
This simplifies to:
$1 + (y + 3)^{2} = 4$
Leading to:
$(y + 3)^{2} = 3$
Taking the square root of both sides:
$y + 3 = ext{±} ext{sqrt}(3)$
Thus,
$y = -3 + ext{sqrt}(3) ext{ and } y = -3 - ext{sqrt}(3).$
The y-coordinates where the circle C crosses the line are therefore:
$y = -3 + ext{sqrt}(3)$ and $y = -3 - ext{sqrt}(3)$ .

the coordinates of the centre of C

the radius of C

the y coordinates of the points where the circle C crosses the line $x = 4$

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