Two circles in the first quadrant of radii r1 and r2 touch the coordinate axes. Each of them cuts of

Q: Why is the center $(r, r)$? What if the circle is not touching the axes?

The problem explicitly states the circle touches both axes in the first quadrant. In the first quadrant, the distance to the x-axis is the y-coordinate, and the distance to the y-axis is the x-coordinate. Since both distances must equal the radius $r$, the center must be $(r, r)$.

Q: Why does the intercept formula $L = 2\sqrt{r^2 - d^2}$ work? Can't I just find the intersection points?

You *could* find the intersection points using the quadratic formula, but that is a long, messy path. The geometric approach using the perpendicular distance is the 'JEE way'—it's elegant, faster, and less prone to calculation errors. It relies on the Pythagorean theorem in the triangle formed by the radius, the perpendicular distance, and half the chord length.

Q: Wait, why do we get two values for $r$? Does that mean there are two different circles?

Exactly! The quadratic equation $r^2 - 4r + 3 = 0$ yields two distinct roots, $r=1$ and $r=3$. This confirms there are indeed two distinct circles in the first quadrant that satisfy the given condition.

Question

Two circles in the first quadrant of radii $r_1$ and $r_2$ touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line $x + y = 2$. Then $r_1^2 + r_2^2 - r_1r_2$ is equal to ____.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on the coordinate plane, looking at the first quadrant. You see a blue line, $x + y = 2$, cutting across the axes.

We are tasked with finding two circles that touch both the $x$-axis and the $y$-axis and cut an intercept of exactly $2$ units on this line. This is a beautiful exercise in symmetry and geometric intuition.

### The First Quadrant Constraint

The first step is to define our circles. If a circle touches both axes in the first quadrant, its center must be equidistant from both axes.

If the radius is $r$, the center must be at $(r, r)$. The equation of such a circle is:

$$(x - r)^2 + (y - r)^2 = r^2$$

This is our starting point and our anchor in the coordinate plane.

### The Bridge to the Line

Now, we need to connect this circle to the line $x + y - 2 = 0$. The key is the perpendicular distance $d$ from the center $(r, r)$ to the line.

Using the standard formula $d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}$, we substitute our center $(r, r)$ and the line equation:

$$d = \frac{|r + r - 2|}{\sqrt{1^2 + 1^2}} = \frac{|2r - 2|}{\sqrt{2}} = \sqrt{2}|r - 1|$$

This distance $d$ serves as the bridge between the circle's center and the line.

### The Pythagorean Intercept

The length of an intercept $L$ cut by a circle on a line is governed by the Pythagorean theorem. If you draw a perpendicular from the center to the chord, you create a right-angled triangle where the radius is the hypotenuse, the perpendicular distance $d$ is one leg, and half the chord length ($L/2$) is the other leg.

Thus, $r^2 = d^2 + (L/2)^2$, which rearranges to $L = 2\sqrt{r^2 - d^2}$. We are given $L = 2$, so $2\sqrt{r^2 - d^2} = 2$, which simplifies to:

$$r^2 - d^2 = 1$$

### The Algebraic Resolution

Now, we substitute our expression for $d$ into this relation: $r^2 - (\sqrt{2}|r - 1|)^2 = 1$. Squaring the distance removes the modulus and the square root:

$$r^2 - 2(r - 1)^2 = 1$$

Expanding this, we get $r^2 - 2(r^2 - 2r + 1) = 1$, which simplifies to $r^2 - 2r^2 + 4r - 2 = 1$. Rearranging the terms, we arrive at the quadratic equation:

$$r^2 - 4r + 3 = 0$$

Factoring this, we get $(r - 1)(r - 3) = 0$, giving us two radii: $r_1 = 1$ and $r_2 = 3$.

### Final Calculation

We have found our two circles. The final step is to evaluate the expression $r_1^2 + r_2^2 - r_1r_2$.

Substituting $r_1 = 1$ and $r_2 = 3$:

$$1^2 + 3^2 - (1)(3) = 1 + 9 - 3 = 7$$

The final result is **7**.

Animated Solution for Mathematics - Circles: Two circles in the first quadrant of radii $r_{1}$ and $r_{2}$ touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line $x + y = 2$ . Then $r_{1}^{2} + r_{2}^{2} - r_{1} r_{2}$ is equal to ____.

Enter Numerical Value:

Visualized Solution

Visualizing the Problem

General Equation of the Circle

Distance from Center to Line

Calculating the Distance

The Intercept Formula

Simplifying the Intercept Equation

Substituting the Distance

Expanding the Equation

Forming the Quadratic

Solving for Radii

Final Calculation Setup

The Final Answer

The Sigma Insight: Intercepts Made by a Circle

Analyzing the Setup

The First Quadrant Constraint

The Bridge to the Line

The Pythagorean Intercept

The Algebraic Resolution

Final Calculation

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