Question

The equation of the circle whose diameter is the common chord of the circles $x^2 + y^2 - 6x - 7 = 0$ and $x^2 + y^2 - 10x + 16 = 0$ is:

• A. $8x^2 + 8y^2 - 92x + 197 = 0$
• B. $x^2 + y^2 - 23x + 197 = 0$
• C. $x^2 + y^2 - \frac{23}{2}x + \frac{197}{4} = 0$
• D. $4x^2 + 4y^2 - 46x + 197 = 0$

Hint:

To find the equation of the circle with the common chord as the diameter, find the points of intersection of the given circles and then use the midpoint and radius formula to construct the new circle's equation.

Answer 1

The Correct Option is A

Step 1: Identify the equations of the given circles: $$C_1: x^2 + y^2 - 6x - 7 = 0, \quad C_2: x^2 + y^2 - 10x + 16 = 0.$$ Step 2: Compute the equation of the common chord (using the radical axis theorem): The radical axis theorem states that the common chord of two circles is given by subtracting their equations. Subtract the equation of $C_2$ from the equation of $C_1$: $$(x^2 + y^2 - 6x - 7) - (x^2 + y^2 - 10x + 16) = 0$$ Simplify this expression: $$-6x - 7 + 10x - 16 = 0 \quad \Rightarrow \quad 4x - 23 = 0 \quad \Rightarrow \quad 4x = 23 \quad \Rightarrow \quad x = \frac{23}{4}.$$ Step 3: Substitute $x = \frac{23}{4}$ back into one of the circle equations to find $y$. Let’s use the first equation of the circle $C_1$: $$x^2 + y^2 - 6x - 7 = 0.$$ Substitute $x = \frac{23}{4}$: $$\left(\frac{23}{4}\right)^2 + y^2 - 6\left(\frac{23}{4}\right) - 7 = 0$$ $$\frac{529}{16} + y^2 - \frac{138}{4} - 7 = 0 \quad \Rightarrow \quad y^2 = \frac{107}{16}.$$ Thus, $y = \pm \frac{\sqrt{107}}{4}$. Step 4: The points of intersection of the two circles give the endpoints of the diameter of the required circle. Use the midpoint formula to find the center of the circle. The midpoint of the points $\left(\frac{23}{4}, \frac{\sqrt{107}}{4}\right)$ and $\left(\frac{23}{4}, -\frac{\sqrt{107}}{4}\right)$ is: $$\left( \frac{23}{4}, 0 \right).$$ Step 5: The radius of the required circle is half the length of the diameter. The distance between the points $\left(\frac{23}{4}, \frac{\sqrt{107}}{4}\right)$ and $\left(\frac{23}{4}, -\frac{\sqrt{107}}{4}\right)$ is: $${Radius} = \frac{\sqrt{107}}{2}.$$ The equation of the circle is: $$\left(x - \frac{23}{4}\right)^2 + y^2 = \left(\frac{\sqrt{107}}{2}\right)^2.$$ Expanding and simplifying: $$\left(x - \frac{23}{4}\right)^2 + y^2 = \frac{107}{4} \quad \Rightarrow \quad \left(x^2 - 2 \times \frac{23}{4}x + \frac{529}{16}\right) + y^2 = \frac{107}{4}.$$ Multiplying the whole equation by 16 to eliminate the denominators: $$16x^2 - 2 \times 23 \times 4x + 529 + 16y^2 = 428.$$ Simplifying further: $$16x^2 - 92x + 16y^2 + 529 = 428 \quad \Rightarrow \quad 16x^2 + 16y^2 - 92x + 101 = 0.$$ Dividing the entire equation by 2: $$8x^2 + 8y^2 - 92x + 197 = 0.$$ Thus, the required equation of the circle is: $$\boxed{8x^2 + 8y^2 - 92x + 197 = 0}.$$