Question

If the radius of the circle x²+y²+ax+by+3=0 is 2, then the point (a, b) lies on the circle

• A. x²+y²=7
• B. x²+y²=4
• C. x²+y²=14
• D. x²+y²=28
• E. x²+y²=1

Answer 1

The Correct Option is D

The general equation of a circle is $x^2 + y^2 + ax + by + c = 0$, where $(h, k)$ is the center and $r$ is the radius. We are given the equation of the circle as $x^2 + y^2 + ax + by + 3 = 0$. The radius of the circle is given as 2, so we use the formula for the radius of a circle: $$r = \sqrt{h^2 + k^2 - c},$$ where $h = -\frac{a}{2}$, $k = -\frac{b}{2}$, and $c = 3$. Substitute the values for $h$ and $k$ into the formula and simplify: $$r = \sqrt{\left( \frac{a}{2} \right)^2 + \left( \frac{b}{2} \right)^2 - 3}.$$ We know that the radius is 2, so we have: $$2 = \sqrt{\left( \frac{a}{2} \right)^2 + \left( \frac{b}{2} \right)^2 - 3}.$$ Squaring both sides: $$4 = \left( \frac{a}{2} \right)^2 + \left( \frac{b}{2} \right)^2 - 3.$$ Simplifying: $$7 = \left( \frac{a}{2} \right)^2 + \left( \frac{b}{2} \right)^2.$$ Thus, the point $(a, b)$ lies on the circle given by the equation $x^2 + y^2 = 28$.

The correct option is (D) : $x^2+y^2=28$