Q103 / 180 · COMEDK 2024 (Morning Shift) · Mathematics
The equation of a circle passing through the origin is $x^2+y^2-6 x+2 y=0$. The equation of one of its diameter is
- A$x-3 y=0$
- Correct answerB$x+3 y=0$
- C$x+y=0$
- D$3 x-y=0$
Solution
The given equation of the circle is $x^2 + y^2 - 6x + 2y = 0$.
Comparing this with the general equation of a circle $x^2 + y^2 + 2gx + 2fy + c = 0$, we have $2g = -6$ and $2f = 2$.
Thus, $g = -3$ and $f = 1$.
The center of the circle is $(-g, -f) = (3, -1)$.
A diameter of a circle is a line passing through its center. Therefore, the equation of the diameter must satisfy the coordinates of the center $(3, -1)$.
Testing the options:
For option (1): $x - 3y = 3 - 3(-1) = 3 + 3 = 6 \neq 0$.
For option (2): $x + 3y = 3 + 3(-1) = 3 - 3 = 0$. This satisfies the equation.
For option (3): $x + y = 3 + (-1) = 2 \neq 0$.
For option (4): $3x - y = 3(3) - (-1) = 9 + 1 = 10 \neq 0$.
Thus, the equation of the diameter is $x + 3y = 0$.
Answer: $x+3 y=0$
Comparing this with the general equation of a circle $x^2 + y^2 + 2gx + 2fy + c = 0$, we have $2g = -6$ and $2f = 2$.
Thus, $g = -3$ and $f = 1$.
The center of the circle is $(-g, -f) = (3, -1)$.
A diameter of a circle is a line passing through its center. Therefore, the equation of the diameter must satisfy the coordinates of the center $(3, -1)$.
Testing the options:
For option (1): $x - 3y = 3 - 3(-1) = 3 + 3 = 6 \neq 0$.
For option (2): $x + 3y = 3 + 3(-1) = 3 - 3 = 0$. This satisfies the equation.
For option (3): $x + y = 3 + (-1) = 2 \neq 0$.
For option (4): $3x - y = 3(3) - (-1) = 9 + 1 = 10 \neq 0$.
Thus, the equation of the diameter is $x + 3y = 0$.
Answer: $x+3 y=0$
Hence, the correct answer is option: 'B'.