Question Statement

Find the equation of the circle passing through the intersection of the circles: $C_1:x^2+y^2-8x-2y+7=0$ $C_2:x^2+y^2-4x+10y+8=0$ and passing through the point $(-1,-2)$.

Hint: The equation of the required circle can be represented as $C_1+\lambda C_2=0$.

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Background and Explanation

To find a circle passing through the intersection of two given circles $S_1=0$ and $S_2=0$, we use the concept of a "Family of Circles." The general equation for any such circle is $S_1+\lambda S_2=0$, where $\lambda$ is a constant determined by an additional condition (in this case, a specific point the circle must pass through).

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Solution

Let the equation of the required circle be represented by the family of circles formula:

$(x^2+y^2-8x-2y+7)+\lambda (x^2+y^2-4x+10y+8)=0$

Finding the value of $\lambda$

Since the required circle passes through the point $(-1,-2)$, we substitute $x=-1$ and $y=-2$ into equation (i):

$\begin{array}{rl}& [(-1{)}^2+(-2{)}^2-8(-1)-2(-2)+7]+\lambda [(-1{)}^2+(-2{)}^2-4(-1)+10(-2)+8]=0\\ & (1+4+8+4+7)+\lambda (1+4+4-20+8)=0\\ & 24+\lambda (-3)=0\\ & 24-3\lambda =0\\ & 3\lambda =24\\ & \lambda =8\end{array}$

Determining the final equation

Now, substitute $\lambda =8$ back into equation (i) to find the specific equation of the circle:

$\begin{array}{rl}& (x^2+y^2-8x-2y+7)+8(x^2+y^2-4x+10y+8)=0\\ & x^2+y^2-8x-2y+7+8x^2+8y^2-32x+80y+64=0\end{array}$

Grouping the like terms together: $(x^2+8x^2)+(y^2+8y^2)+(-8x-32x)+(-2y+80y)+(7+64)=0$

$9x^2+9y^2-40x+78y+71=0$

This is the required equation of the circle.

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Key Formulas or Methods Used

• Family of Circles: The equation of a circle passing through the intersection of two circles $C_1=0$ and $C_2=0$ is $C_1+\lambda C_2=0$.
• Point Substitution: Substituting a known point $(x,y)$ into an equation to solve for an unknown parameter ($\lambda$).
• Algebraic Simplification: Combining like terms to reach the standard form of a circle's equation.

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Summary of Steps

1. Set up the general equation of the family of circles using $C_1+\lambda C_2=0$.
2. Substitute the coordinates of the given point $(-1,-2)$ into the equation.
3. Solve the resulting linear equation to find the value of the constant $\lambda$.
4. Substitute the value of $\lambda$ back into the family equation.
5. Expand the brackets and simplify by combining like terms to get the final equation.