Key Concepts and Formulas

• Equation of a Circle Touching Coordinate Axes: A circle in the first quadrant touching both the x-axis and y-axis has its center at $(r, r)$ and radius $r$. Its equation is $(x-r)^2 + (y-r)^2 = r^2$.
• Perpendicular Distance from a Point to a Line: The perpendicular distance $d$ from a point $(x_0, y_0)$ to a line $ax + by + c = 0$ is given by $d = \frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}}$.
• Length of Intercept by a Circle on a Line: If a circle of radius $r$ cuts a line, and the perpendicular distance from the center of the circle to the line is $d$, then the length of the intercept is $2\sqrt{r^2 - d^2}$.

Step-by-Step Solution

Step 1: Write the equation of the circles.

Since both circles touch the coordinate axes in the first quadrant, their centers are at $(r_1, r_1)$ and $(r_2, r_2)$, and their equations are: $(x - r_1)^2 + (y - r_1)^2 = r_1^2$ $(x - r_2)^2 + (y - r_2)^2 = r_2^2$

Step 2: Calculate the perpendicular distance from the center of each circle to the line $x + y = 2$.

The equation of the line can be written as $x + y - 2 = 0$. The perpendicular distance $d_1$ from $(r_1, r_1)$ to the line is: $d_1 = \frac{|r_1 + r_1 - 2|}{\sqrt{1^2 + 1^2}} = \frac{|2r_1 - 2|}{\sqrt{2}} = \frac{|2(r_1 - 1)|}{\sqrt{2}} = \sqrt{2}|r_1 - 1|$ Similarly, the perpendicular distance $d_2$ from $(r_2, r_2)$ to the line is: $d_2 = \frac{|r_2 + r_2 - 2|}{\sqrt{1^2 + 1^2}} = \frac{|2r_2 - 2|}{\sqrt{2}} = \frac{|2(r_2 - 1)|}{\sqrt{2}} = \sqrt{2}|r_2 - 1|$

Step 3: Use the intercept length information.

The length of the intercept cut off by each circle on the line $x + y = 2$ is 2. Using the formula for the intercept length, we have: $2\sqrt{r_1^2 - d_1^2} = 2$ $2\sqrt{r_2^2 - d_2^2} = 2$ Squaring both equations, we get: $r_1^2 - d_1^2 = 1$ $r_2^2 - d_2^2 = 1$

Step 4: Substitute the expressions for $d_1$ and $d_2$.

Substituting the expressions for $d_1$ and $d_2$ from Step 2 into the equations from Step 3, we have: $r_1^2 - (\sqrt{2}|r_1 - 1|)^2 = 1$ $r_2^2 - (\sqrt{2}|r_2 - 1|)^2 = 1$ Simplifying, we get: $r_1^2 - 2(r_1 - 1)^2 = 1$ $r_2^2 - 2(r_2 - 1)^2 = 1$ Expanding and simplifying further: $r_1^2 - 2(r_1^2 - 2r_1 + 1) = 1$ $r_2^2 - 2(r_2^2 - 2r_2 + 1) = 1$ $-r_1^2 + 4r_1 - 2 = 1$ $-r_2^2 + 4r_2 - 2 = 1$ So, we have two quadratic equations: $r_1^2 - 4r_1 + 3 = 0$ $r_2^2 - 4r_2 + 3 = 0$

Step 5: Solve the quadratic equations.

Both quadratic equations are the same, so $r_1$ and $r_2$ are the roots of the equation $r^2 - 4r + 3 = 0$. Factoring the quadratic, we have: $(r - 1)(r - 3) = 0$ Thus, the roots are $r = 1$ and $r = 3$. Therefore, $r_1 = 1$ and $r_2 = 3$ (or vice versa).

Step 6: Calculate the required expression.

We need to find the value of $r_1^2 + r_2^2 - r_1r_2$. Substituting $r_1 = 1$ and $r_2 = 3$, we get: $r_1^2 + r_2^2 - r_1r_2 = (1)^2 + (3)^2 - (1)(3) = 1 + 9 - 3 = 7$ However, the correct answer is 2. Let's re-examine the problem. The intercept equation is $2\sqrt{r^2 - d^2} = 2$, which implies $r^2 - d^2 = 1$. Substituting $d = \sqrt{2}|r-1|$, we have $r^2 - 2(r-1)^2 = 1$, which simplifies to $r^2 - 4r + 3 = 0$. The roots are $r=1, 3$. So $r_1 = 1$ and $r_2 = 3$. The expression we need to evaluate is $r_1^2 + r_2^2 - r_1 r_2 = 1^2 + 3^2 - (1)(3) = 1 + 9 - 3 = 7$. There's been an error.

Let's consider the case where the intercept is 2. $r^2 - 2(r-1)^2 = 1$, which gives $r^2 - 2r^2 + 4r - 2 = 1$, or $r^2 - 4r + 3 = 0$. Then $(r-1)(r-3) = 0$ and $r_1 = 1, r_2 = 3$.

Now, consider the case where the line $x+y=2$ intersects the circle. The equation of the circle is $(x-r)^2+(y-r)^2=r^2$. The distance from the center to the line is $d = \frac{|r+r-2|}{\sqrt{2}} = \frac{|2r-2|}{\sqrt{2}} = \sqrt{2}|r-1|$. We have $r^2 - d^2 = 1$, so $r^2 - 2(r-1)^2 = 1$, which implies $r^2 - 2(r^2-2r+1) = 1$, so $-r^2 + 4r - 2 = 1$, or $r^2-4r+3 = 0$. So $r = 1, 3$.

$r_1 = 1, r_2 = 3$. We want $r_1^2 + r_2^2 - r_1 r_2 = 1^2 + 3^2 - (1)(3) = 1 + 9 - 3 = 7$.

The given correct answer is 2. Something is wrong. Let's work backwards. If $r_1^2 + r_2^2 - r_1 r_2 = 2$, then $r_1^2 + r_2^2 - r_1 r_2 = (r_1+r_2)^2 - 3r_1r_2 = 2$. Also, $r_1+r_2 = 4$ and $r_1 r_2 = 3$. Then $16 - 3(3) = 7$, not 2.

There must be an error with the question or the answer. Let us assume the question is correct and try a different approach. If the intercept is 2, then $4 = 4(r^2 - d^2)$, so $1 = r^2 - d^2$. Thus $1 = r^2 - 2(r-1)^2$, which means $r^2 - 4r + 3 = 0$. Then $r = 1, 3$. So $r_1 = 1, r_2 = 3$. Then $r_1^2 + r_2^2 - r_1 r_2 = 1 + 9 - 3 = 7$.

Common Mistakes & Tips

• Be careful with absolute values when calculating distances.
• Double-check your algebraic manipulations to avoid errors.
• Remember the formula for the length of an intercept.

Summary

We found the radii of the two circles to be 1 and 3. Substituting these values into the expression $r_1^2 + r_2^2 - r_1 r_2$, we get $1^2 + 3^2 - (1)(3) = 1 + 9 - 3 = 7$. However, the correct answer provided is 2. There appears to be an error in the given correct answer. Based on the problem statement, the value of the expression is 7.

Final Answer The final answer is \boxed{7}. There seems to be an error in the provided "Correct Answer" as the correct value is 7.