JEE Advanced 1992

LEVELJEE Advanced

Animated Solution for Mathematics - Circles: Three circles touch one another externally. The tangent at their point of contact meet at a point whose distance from a point of contact is $4$ . Find the ratio of the product of the radii to the sum of the radii of the circles.

Q: Does this result depend on the circles being of different sizes?

Not at all! The derivation $r^2 = \frac{r_1 r_2 r_3}{r_1 + r_2 + r_3}$ holds true regardless of the specific values of the radii. The beauty of this problem lies in its generality; the ratio remains constant as long as the circles touch externally and the tangents meet at a point.

Progressive Hints

Hint 1

Hint 1: Visualize the centers of the three circles as vertices of a triangle. When two circles touch externally, the distance between their centers is the sum of their radii. Can you express the side lengths of this triangle in terms of

r_{1}, r_{2},

and

r_{3}

Hint 2

Hint 2: Consider the common tangents at the points of contact. These tangents are perpendicular to the line connecting the centers of the circles. Since all three tangents meet at a single point

O

, this point

O

must be equidistant from the sides of the triangle formed by the centers. What special point in a triangle is equidistant from its sides?

Hint 3

Hint 3: You have identified

O

as the incenter and the distance

4

as the inradius

r

. Use the fundamental relationship

r = \frac{Δ}{s}

, where

Δ

is the area of the triangle and

s

is the semi-perimeter. Apply Heron's formula to express

Δ

in terms of

r_{1}, r_{2},

and

r_{3}

Visualized Solution

Visualizing the Three Circles

Let the three circles have centers $A, B, C$ and radii $r_{1}, r_{2}, r_{3}$ .
They touch each other externally at points $P, Q, R$ .

Forming Triangle $A B C$

Join the centers to form $Δ A B C$ .
The sides pass through the points of contact.
Side lengths: $A B = r_{1} + r_{2}$ , $B C = r_{2} + r_{3}$ , $C A = r_{3} + r_{1}$ .

The Common Tangents

Draw common tangents at the points of contact $P, Q, R$ .
These tangents meet at a single point $O$ .
We are given that the distance from $O$ to any point of contact is $4$ .
So, $O P = O Q = O R = 4$ .

Identifying Point $O$

The common tangent at $P$ is perpendicular to the line of centers $A B$ .
Therefore, $O P ⊥ A B$ , $O Q ⊥ B C$ , and $O R ⊥ C A$ .
Since $O$ is equidistant from the sides of $Δ A B C$ , $O$ is the incenter.
The distance $4$ is the inradius ( $r$ ) of $Δ A B C$ .

Semi-perimeter of $Δ A B C$

Let's find the semi-perimeter $s$ of $Δ A B C$ .
$s = \frac{A B + B C + C A}{2}$
$s = \frac{( r _{1} + r _{2} ) + ( r _{2} + r _{3} ) + ( r _{3} + r _{1} )}{2}$
$s = r_{1} + r_{2} + r_{3}$

Area of $Δ A B C$

Using Heron's formula: $Δ = s (s - a) (s - b) (s - c)$
Here, $a = r_{2} + r_{3}$ , $b = r_{3} + r_{1}$ , $c = r_{1} + r_{2}$ .
$s - a = (r_{1} + r_{2} + r_{3}) - (r_{2} + r_{3}) = r_{1}$
Similarly, $s - b = r_{2}$ and $s - c = r_{3}$ .

Substituting into Heron's Formula

Substitute these into the area formula.
$Δ = (r_{1} + r_{2} + r_{3}) r_{1} r_{2} r_{3}$
This gives us the area of the triangle in terms of the radii.

The Inradius Formula

We know the relation between area $Δ$ , semi-perimeter $s$ , and inradius $r$ .
$r = \frac{Δ}{s}$
We have $r = 4$ , $Δ = (r_{1} + r_{2} + r_{3}) r_{1} r_{2} r_{3}$ , and $s = r_{1} + r_{2} + r_{3}$ .

Equating the Inradius

Substitute the expressions into $r = \frac{Δ}{s}$ :
$4 = \frac{( r _{1} + r _{2} + r _{3} ) r _{1} r _{2} r _{3}}{r _{1} + r _{2} + r _{3}}$
Simplify the denominator by bringing it inside the square root.

Simplifying the Expression

$4 = \frac{( r _{1} + r _{2} + r _{3} ) r _{1} r _{2} r _{3}}{( r _{1} + r _{2} + r _{3} ) ^{2}}$
Cancel the common term $(r_{1} + r_{2} + r_{3})$ .
$4 = \frac{r _{1} r _{2} r _{3}}{r _{1} + r _{2} + r _{3}}$

Final Ratio Calculation

Square both sides to remove the square root.
$16 = \frac{r _{1} r _{2} r _{3}}{r _{1} + r _{2} + r _{3}}$
The numerator is the product of the radii, and the denominator is the sum.
The ratio of the product to the sum is $16 : 1$ .

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The Sigma Insight: Equation of Tangent and Normal

Solution Diagram

Analyzing the Setup

Imagine three circles with centers

A, B,

and

C

and radii

r_{1}, r_{2},

and

r_{3}

. These circles touch each other externally at points

P, Q,

and

R

This configuration creates a triangle

A B C

where the vertices are the centers of the circles. Because the circles touch externally, the side lengths of this triangle are determined by the sum of the radii of the touching circles.

The side lengths are:

A B = r_{1} + r_{2}

B C = r_{2} + r_{3}

C A = r_{3} + r_{1}

The Incenter Revelation

Consider the common tangents at the points of contact

P, Q,

and

R

. These tangents intersect at a single point

O

We are given that the distance from

O

to any point of contact is

4

. Since these tangents are perpendicular to the sides of

Δ A B C

, the point

O

is equidistant from the sides of the triangle.

Therefore,

O

is the incenter of

Δ A B C

, and the distance

4

is the inradius

r

. This realization is the key to unlocking the relationship between the radii.

The Power of Heron's Formula

To relate the radii to the inradius, we first calculate the semi-perimeter

s

Δ A B C

s = \frac{( r _{1} + r _{2} ) + ( r _{2} + r _{3} ) + ( r _{3} + r _{1} )}{2} = r_{1} + r_{2} + r_{3}

Next, we apply Heron's formula for the area

Δ

of the triangle:

Δ = s (s - a) (s - b) (s - c)

Substituting the side lengths

a = r_{2} + r_{3}

b = r_{3} + r_{1}

, and

c = r_{1} + r_{2}

, we find:

s - a = r_{1}, s - b = r_{2}, s - c = r_{3}

Thus, the area simplifies to:

Δ = (r_{1} + r_{2} + r_{3}) r_{1} r_{2} r_{3}

The Final Synthesis

We utilize the fundamental relationship between the inradius

r

, the area

Δ

, and the semi-perimeter

s

r = \frac{Δ}{s}

Substituting our known values into this equation:

4 = \frac{( r _{1} + r _{2} + r _{3} ) r _{1} r _{2} r _{3}}{r _{1} + r _{2} + r _{3}}

By bringing the denominator inside the square root, we obtain:

4 = \frac{r _{1} r _{2} r _{3}}{r _{1} + r _{2} + r _{3}}

Squaring both sides, we arrive at the final result:

\frac{r _{1} r _{2} r _{3}}{r _{1} + r _{2} + r _{3}} = 16

The ratio of the product of the radii to their sum is exactly 16.

Common Traps & FAQs

Wait, why is the intersection point of the tangents the incenter?

Great question! The common tangent at the point of contact between two circles is perpendicular to the line joining their centers. Since the tangents meet at point

O

, and each tangent is perpendicular to a side of the triangle formed by the centers,

O

is equidistant from all three sides. By definition, the point equidistant from the sides of a triangle is the incenter.

Can I use coordinate geometry to solve this instead of Heron's formula?

You could, but it would be significantly more complex. You would need to set up a coordinate system, find the equations of the lines, and then find the intersection point. Using the geometric properties of the incenter and Heron's formula is the elegant, intended path for this problem.

Does this result depend on the circles being of different sizes?

Not at all! The derivation

r^{2} = \frac{r _{1} r _{2} r _{3}}{r _{1} + r _{2} + r _{3}}

holds true regardless of the specific values of the radii. The beauty of this problem lies in its generality; the ratio remains constant as long as the circles touch externally and the tangents meet at a point.

Animated Solution for Mathematics - Circles: Three circles touch one another externally. The tangent at their point of contact meet at a point whose distance from a point of contact is $4$ . Find the ratio of the product of the radii to the sum of the radii of the circles.

Visualized Solution

Visualizing the Three Circles

Forming Triangle $A B C$

The Common Tangents

Identifying Point $O$

Semi-perimeter of $Δ A B C$

Area of $Δ A B C$

Substituting into Heron's Formula

The Inradius Formula

Equating the Inradius

Simplifying the Expression

Final Ratio Calculation

The Sigma Insight: Equation of Tangent and Normal

Analyzing the Setup

The Incenter Revelation

The Power of Heron's Formula

The Final Synthesis

Similar Questions

Circles with radii 3, 4 and 5 touch each other externally. If $P$ is the point of intersection of tangents to these circles at their points of contact, find the distance of $P$ from the points of contact.

Three circles of radii a, b, c (a < b < c) touch each other externally. If they have x-axis as a common tangent, then :

Two circles with equal radii are intersecting at the points $(0, 1)$ and $(0, - 1)$ . The tangent at the point $(0, 1)$ to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is :

Two circles each of radius 5 units touch each other at the point $(1, 2)$ . If the equation of their common tangent is $4 x + 3 y = 10$ , and $C_{1} (α, β)$ and $C_{2} (γ, δ)$ , $C_{1} \neq = C_{2}$ are their centres, then $∣ (α + β) (γ + δ) ∣$ is equal to .

If a circle C passing through the point (4,0) touches the circle $x^{2} + y^{2} + 4 x - 6 y = 12$ externally at the point (1, -1), then the radius of C is :

Points $A, B$ and $C$ lie on the parabola $y^{2} = 4 a x$ . The tangents to the parabola at $A, B$ and $C$ , taken in pairs, intersect at points $P, Q$ and $R$ . Determine the ratio of the areas of the triangles $A B C$ and $P QR$ .

Find the coordinates of the point at which the circles $x^{2} + y^{2} - 4 x - 2 y - 4 = 0$ and $x^{2} + y^{2} - 12 x - 8 y - 36 = 0$ touch each other. Also find equations common tangents touching the circles in the distinct points.

Visualized Solution

Visualizing the Three Circles

Forming Triangle .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ABC

The Common Tangents

Identifying Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } O

Semi-perimeter of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ΔABC

Area of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ΔABC

Substituting into Heron's Formula

The Inradius Formula

Equating the Inradius

Simplifying the Expression

Final Ratio Calculation

The Sigma Insight: Equation of Tangent and Normal

Analyzing the Setup

The Incenter Revelation

The Power of Heron's Formula

The Final Synthesis

Similar Questions

Three circles of radii a, b, c (a < b < c) touch each other externally. If they have x-axis as a common tangent, then :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a circle C passing through the point (4,0) touches the circle x2+y2+4x−6y=12 externally at the point (1, -1), then the radius of C is :

Forming Triangle $A B C$

Identifying Point $O$

Semi-perimeter of $Δ A B C$

Area of $Δ A B C$

If a circle C passing through the point (4,0) touches the circle $x^{2} + y^{2} + 4 x - 6 y = 12$ externally at the point (1, -1), then the radius of C is :