Circle And System Of Circles Question 172
Question: For $ ax^{2}+2hxy+3y^{2}+4x+8y-6=0 $ to represent a circle, one must have
Options:
A) $ a=3,\ h=0 $
B) $ a=1,\ h=0 $
C) $ a=h=3 $
D) $ a=h=0 $
Show Answer
Answer:
Correct Answer: A
Solution:
The given equation is of a general second degree equation in two variables, which can represent different types of conic sections such as circle, ellipse, hyperbola, or parabola. The equation of a circle can be written in the general form as:
$ ax^{2} + by^{2} + 2hxy + 2gx + 2fy + c = 0 $
For this to represent a circle, the coefficients of $x^{2}$ and $y^{2}$ must be equal (i.e., $a = b$), and the coefficient of the xy term must be zero (i.e., $h = 0$).
In the given equation, the coefficient of $y^{2}$ is 3. So, for the equation to represent a circle, $a$ must also be 3, and $h$ must be 0.
Therefore, the correct answer is Option A: $a=3,\ h=0$. This is not just obvious, but it follows from the standard form of the equation of a circle.
BETA
