The line $x + 3y = 0$ is a diameter of the circle $x^2 + y^2 + 6x + 2y = 0$.

The line x + 3y = 0 is a diameter of the circle x² + y² + 6x + 2y = 0.

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x + 3y = 0

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Concept understanding — Circle Equation Standard Form

Where the Circle Equation Comes From

Imagine you're standing at a point on a flat field. You tie a rope to a stake at that point, walk out the full length of the rope, and start walking in a circle, keeping the rope taut. Every point you step on is exactly the same distance from the stake.

That's the entire idea: a circle is the set of all points that are a fixed distance (the radius) from a fixed point (the centre).

If we put this on a coordinate plane, we can turn that geometric idea into an algebraic equation.

From Geometry to Algebra

Let the centre be at coordinates $(h, k)$ . Let the radius be $r$ . Take any point $(x, y)$ that lies on the circle. The distance from $(x, y)$ to $(h, k)$ must equal $r$ .

What's the distance between two points in the plane? The distance formula:

\sqrt{(x - h)^2 + (y - k)^2} = r

Now square both sides to remove the square root:

(x - h)^2 + (y - k)^2 = r^2

That's it. That's the standard form of the equation of a circle.

(x - h)^2 + (y - k)^2 = r^2

What Each Piece Tells You

$(x - h)$ and $(y - k)$ — these shift the circle away from the origin. If the centre is at $(0,0)$ , the equation simplifies to $x^2 + y^2 = r^2$ .
$r^2$ — notice it's the square of the radius, not the radius itself. If the equation says $x^2 + y^2 = 25$ , the radius is $\sqrt{25} = 5$ , not 25.
The equals sign — only the points $(x, y)$ that make this equation true lie on the circle. Any other point gives a larger or smaller left-hand side.

Watch out

A common mistake: for $(x - 3)^2 + (y + 2)^2 = 16$ , students often read the centre straight off the signs printed in the equation and say $(3, 2)$ . That's wrong. Each bracket must first be written in the exact form $x - h$ and $y - k$ : here $(y + 2) = (y - (-2))$ , so $k = -2$ , not $2$ . The centre is actually $(3, -2)$ . Always flip the sign inside every bracket before reading off $h$ and $k$ .

Quick Example

Write the equation of a circle with centre $(-1, 4)$ and radius $3$ .

Here $h = -1$ , $k = 4$ , $r = 3$ . Plug in:

(x - (-1))^2 + (y - 4)^2 = 3^2

Simplify:

$(x + 1)^2 + (y - 4)^2 = 9$ …

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A diameter of a circle must pass through its center. By finding the center of the given circle and substituting its coordinates into the line equation, we find that the center does not lie on the line, meaning the statement is false.

To determine if a given line is a diameter of a circle, we need to understand the fundamental property of a diameter: it is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. This means that if a line is a diameter, it must contain the center of the circle.

Our approach will be to first find the center of the given circle and then check if this center point satisfies the equation of the given line. If it does, the line passes through the center and thus could be a diameter. If it doesn't, the line cannot be a diameter.

Identify the standard form of a circle equation and extract the center.

The general equation of a circle is given by $x^2 + y^2 + 2gx + 2fy + c = 0$ .

For a circle in this form, its center is at the point $(-g, -f)$ .

The given circle equation is $x^2 + y^2 + 6x + 2y = 0$ .

Comparing this to the general form:

$2g = 6 \implies g = 3$

$2f = 2 \implies f = 1$

The constant term $c$ is $0$ in this case, but it doesn't affect the center coordinates.

Therefore, the center of the circle is $(-g, -f) = (-3, -1)$ .
$x^2 + y^2 + 2gx + 2fy + c = 0$
Check if the circle's center lies on the given line.

The given line is $x + 3y = 0$ .

For the line to be a diameter, the center of the circle, which is $(-3, -1)$ , must satisfy the equation of the line.

Substitute $x = -3$ and $y = -1$ into the line equation: …

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Q.The line x+3y=0x + 3y = 0x+3y=0 is a diameter of the circle x2+y2+6x+2y=0x^2 + y^2 + 6x + 2y = 0x2+y2+6x+2y=0.

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Q.The line $x + 3y = 0$ is a diameter of the circle $x^2 + y^2 + 6x + 2y = 0$ .

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