SSCE HSC Mathematics Advanced Absolute value functions: The circle $x^2

The circle $x^2 - 6x + y^2 + 4y = 3$ is reflected in the x-axis - HSC - SSCE Mathematics Advanced - Question 24 - 2020 - Paper 1

Question 24

The circle $x^2 - 6x + y^2 + 4y = 3$ is reflected in the x-axis. Sketch the reflected circle, showing the coordinates of the centre and the radius.

Worked Solution & Example Answer:The circle $x^2 - 6x + y^2 + 4y = 3$ is reflected in the x-axis - HSC - SSCE Mathematics Advanced - Question 24 - 2020 - Paper 1

Step 1

96%

114 rated

Answer

To find the center and radius of the original circle, we need to rewrite the equation in standard form:

Starting from the equation: $x^2 - 6x + y^2 + 4y = 3$

Complete the square for the $x$ terms: $x^2 - 6x = (x - 3)^2 - 9$

Complete the square for the $y$ terms: $y^2 + 4y = (y + 2)^2 - 4$

Substituting back into the equation gives us: $(x - 3)^2 - 9 + (y + 2)^2 - 4 = 3$ $(x - 3)^2 + (y + 2)^2 = 16$

So, the original circle has a center at $(3, -2)$ and a radius of $4$ .

Step 2

99%

104 rated

Answer

When reflected in the x-axis, the $y$ -coordinate of the center changes sign:

The radius remains the same at $4$ .

Step 3

96%

101 rated

Answer

To sketch the reflected circle:

Draw the original circle centered at $(3, -2)$ with radius $4$ .
Then draw the reflected circle centered at $(3, 2)$ with the same radius of $4$ . Make sure to label the centers and the radius on your sketch.