JEE Main 2021 (17 March Shift 1)

LEVELJEE Main

Animated Solution for Mathematics - Circles: Choose the incorrect statement about the two circles whose equations are given below: $x^{2} + y^{2} - 10 x - 10 y + 41 = 0$ and $x^{2} + y^{2} - 16 x - 10 y + 80 = 0$

Q: Wait, why do we reject the negative root when calculating the radius?

The radius represents a physical distance, which must always be positive. The formula $\sqrt{g^2 + f^2 - c}$ inherently yields a positive value because the square root function (principal root) is defined as non-negative. In geometry, a negative radius has no physical meaning.

Q: Why can't we just say the centers are inside each other because they are close?

In JEE Advanced, 'close' is not a mathematical proof. We must be rigorous. The definition of 'inside' is $d < r$. If $d = r$, the point is on the boundary. Precision is the difference between a correct answer and a negative mark. Always rely on the inequality, not your intuition.

Q: Is there a faster way to find the center without using the formula?

Yes! You can complete the square. For $x^2 - 10x$, add 25 to make it $(x-5)^2$. It is a great way to verify your results, but the formula $(-g, -f)$ is your most reliable tool under exam pressure. Use the formula for speed, and completing the square for verification.

Select Answer:

Progressive Hints

Hint 1

Hint 1 (The Spark): Before diving into the algebra, visualize the two circles. They are not just equations; they are geometric entities. Ask yourself: what does it mean for a circle to pass through the center of another? It implies a specific relationship between the distance between their centers (d) and their radii (r1, r2).

Hint 2

Hint 2 (The Arsenal): Your toolkit for this problem is the General Equation of a Circle:

x^{2} + y^{2} + 2 g x + 2 f y + c = 0

. Extract the centers (h, k) and radii r for both circles. Once you have these, the problem transforms from algebra into a simple comparison of distances.

Hint 3

Hint 3 (The Execution): Calculate the distance d between the centers (5, 5) and (8, 5). Now, compare this d with the radii r1 and r2. Remember the strict definition: a point is 'inside' a circle only if its distance from the center is strictly less than the radius (d < r). If d = r, the point lies on the circumference.

Visualized Solution

Introduction to the Problem

Objective: Identify the incorrect statement among the given options.
Circle 1 ( $C_{1}$ ): $x^{2} + y^{2} - 10 x - 10 y + 41 = 0$
Circle 2 ( $C_{2}$ ): $x^{2} + y^{2} - 16 x - 10 y + 80 = 0$

The General Formula

General Form: $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$
Center Formula: $(- g, - f)$
Radius Formula: $r = g^{2} + f^{2} - c$

Center of Circle $C_{1}$

For $C_{1}$ : $x^{2} + y^{2} - 10 x - 10 y + 41 = 0$
$2 g = - 10 ⟹ g = - 5$
$2 f = - 10 ⟹ f = - 5$
Center $c_{1}$ : $(5, 5)$

Radius of Circle $C_{1}$

Radius of $C_{1}$ : $r_{1} = g^{2} + f^{2} - c$
$r_{1} = (- 5)^{2} + (- 5)^{2} - 41$
$r_{1} = 25 + 25 - 41 = 9$
Result: $r_{1} = 3$

Center of Circle $C_{2}$

For $C_{2}$ : $x^{2} + y^{2} - 16 x - 10 y + 80 = 0$
$2 g = - 16 ⟹ g = - 8$
$2 f = - 10 ⟹ f = - 5$
Center $c_{2}$ : $(8, 5)$

Radius of Circle $C_{2}$

Radius of $C_{2}$ : $r_{2} = g^{2} + f^{2} - c$
$r_{2} = (- 8)^{2} + (- 5)^{2} - 80$
$r_{2} = 64 + 25 - 80 = 9$
Result: $r_{2} = 3$

Distance Between Centers

Distance Formula: $d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
$d = (8 - 5)^{2} + (5 - 5)^{2}$
$d = 3^{2} + 0^{2} = 9$
Result: $d = 3$

Evaluating Option 1

Option 1: Distance is the average of radii.
Average of Radii: $\frac{r _{1} + r _{2}}{2} = \frac{3 + 3}{2} = 3$
Comparison: $d = 3$ and Average $= 3$
Conclusion: Statement 1 is Correct.

Evaluating Option 3

Option 3: Both circles pass through the center of each other.
Condition: If $d = r_{1}$ and $d = r_{2}$ , each circle passes through the other's center.
Verification: $d = 3$ , $r_{1} = 3$ , $r_{2} = 3$
Conclusion: Statement 3 is Correct.

Evaluating Option 4

Option 4: Circles have two intersection points.
Condition for 2 points: $∣ r_{1} - r_{2} ∣ < d < r_{1} + r_{2}$
Verification: $∣3 - 3∣ < 3 < 3 + 3 ⟹ 0 < 3 < 6$
Conclusion: Statement 4 is Correct.

Evaluating Option 2 (The Trap)

Option 2: Both circles' centers lie inside region of one another.
Condition for 'Inside': Distance $d$ must be strictly less than radius $r$ ( $d < r$ ).
Our Case: $d = 3$ and $r = 3 ⟹ d = r$

Final Conclusion

Reality: Since $d = r$ , the centers lie exactly on the circumference, not strictly inside.
Conclusion: Statement 2 is Incorrect.
Final Answer: Option (2)

00:00 / 00:00

The Sigma Insight: Position of a Point with Respect to a Circle

Solution Diagram

Decoding the Circle Equations

When you encounter equations like

x^{2} + y^{2} - 10 x - 10 y + 41 = 0

, view them as the fingerprints of geometric shapes. By comparing these to the general form

x^{2} + y^{2} + 2 g x + 2 f y + c = 0

, we can extract the center

(- g, - f)

and the radius

r = g^{2} + f^{2} - c

For the first circle,

C_{1}

, we identify the center at

(5, 5)

and calculate the radius:

r_{1} = 5^{2} + 5^{2} - 41 = 25 + 25 - 41 = 9 = 3

For the second circle,

C_{2}

, we perform the same analysis to find the center at

(8, 5)

and a radius of:

r_{2} = 3

Analyzing the Geometric Relationship

Now, we calculate the distance

d

between the two centers

(5, 5)

and

(8, 5)

using the distance formula:

d = (8 - 5)^{2} + (5 - 5)^{2} = 3^{2} + 0^{2} = 3

This is the pivotal moment in our analysis. We have two circles, both with a radius of

3

, separated by a distance of

3

The Trap of the Boundary

Because the distance between the centers is exactly equal to the radius (

d = r

), the center of one circle lies exactly on the circumference of the other.

Many students mistakenly assume that if the distance is equal to the radius, the center is "inside." However, mathematically, the condition for a center to be strictly inside is

d < r

Since

d = r

, the center is located precisely on the boundary. This distinction is the difference between a correct answer and a common pitfall. Keep this geometric precision in mind, and you will successfully navigate any JEE coordinate geometry problem.

Common Traps & FAQs

Wait, why do we reject the negative root when calculating the radius?

The radius represents a physical distance, which must always be positive. The formula

g^{2} + f^{2} - c

inherently yields a positive value because the square root function (principal root) is defined as non-negative. In geometry, a negative radius has no physical meaning.

Why can't we just say the centers are inside each other because they are close?

In JEE Advanced, 'close' is not a mathematical proof. We must be rigorous. The definition of 'inside' is

d < r

. If

d = r

, the point is on the boundary. Precision is the difference between a correct answer and a negative mark. Always rely on the inequality, not your intuition.

Is there a faster way to find the center without using the formula?

Yes! You can complete the square. For

x^{2} - 10 x

, add 25 to make it

(x - 5)^{2}

. It is a great way to verify your results, but the formula

(- g, - f)

is your most reliable tool under exam pressure. Use the formula for speed, and completing the square for verification.

Animated Solution for Mathematics - Circles: Choose the incorrect statement about the two circles whose equations are given below: $x^{2} + y^{2} - 10 x - 10 y + 41 = 0$ and $x^{2} + y^{2} - 16 x - 10 y + 80 = 0$

Select Answer:

Visualized Solution

Introduction to the Problem

The General Formula

Center of Circle $C_{1}$

Radius of Circle $C_{1}$

Center of Circle $C_{2}$

Radius of Circle $C_{2}$

Distance Between Centers

Evaluating Option 1

Evaluating Option 3

Evaluating Option 4

Evaluating Option 2 (The Trap)

Final Conclusion

The Sigma Insight: Position of a Point with Respect to a Circle

Decoding the Circle Equations

Analyzing the Geometric Relationship

The Trap of the Boundary

Similar Questions

Let $r_{1}$ and $r_{2}$ be the radii of the largest and smallest circles, respectively, which pass through the point $(- 4, 1)$ and having their centres on the circumference of the circle $x^{2} + y^{2} + 2 x + 4 y - 4 = 0$ . If $\frac{r _{1}}{r _{2}} = a + b 2$ , then $a + b$ is equal to :

Let the circle $S : 36 x^{2} + 36 y^{2} - 108 x + 120 y + C = 0$ be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, $x - 2 y = 4$ and $2 x - y = 5$ lies inside the circle $S$ , then :

If a variable line, $3 x + 4 y - λ = 0$ is such that the two circles $x^{2} + y^{2} - 2 x - 2 y + 1 = 0$ and $x^{2} + y^{2} - 18 x - 2 y + 78 = 0$ are on its opposite sides, then the set of all values of $λ$ is the interval :-

Comprehension Passage

Consider $M$ with $r = \frac{1025}{513}$ . Let $k$ be the number of all those circles $C_{n}$ that are inside $M$ . Let $l$ be the maximum possible number of circles among these $k$ circles such that no two circles intersect. Then

Consider $M$ with $r = \frac{( 2 ^{199} - 1 ) 2}{2 ^{198}}$ . The number of all those circles $D_{n}$ that are inside $M$ is

The straight line $2 x - 3 y = 1$ divides the circular region $x^{2} + y^{2} \leq 6$ into two parts. If $S = {(2, 3/4), (5/2, 3/4), (1/4, - 1/4), (1/8, 1/4)}$ then the number of points (s) in $S$ lying inside the smaller part is

The set of values of $k$ for which the circle $C : 4 x^{2} + 4 y^{2} - 12 x + 8 y + k = 0$ lies inside the fourth quadrant and the point $(1, - \frac{1}{3})$ lies on or inside the circle $C$ is :

Let the maximum and minimum values of $(8 x - x^{2} - 12 - 4)^{2} + (x - 7)^{2}, x \in R$ be $M$ and $m$ , respectively. Then $M^{2} - m^{2}$ is equal to _________

Select Answer:

Visualized Solution

Introduction to the Problem

The General Formula

Center of Circle .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C1​

Radius of Circle .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C1​

Center of Circle .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C2​

Radius of Circle .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C2​

Distance Between Centers

Evaluating Option 1

Evaluating Option 3

Evaluating Option 4

Evaluating Option 2 (The Trap)

Final Conclusion

The Sigma Insight: Position of a Point with Respect to a Circle

Decoding the Circle Equations

Analyzing the Geometric Relationship

The Trap of the Boundary

Similar Questions

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Consider M with r=2198(2199−1)2​​. The number of all those circles Dn​ that are inside M is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The set of values of k for which the circle C:4x2+4y2−12x+8y+k=0 lies inside the fourth quadrant and the point (1,−31​) lies on or inside the circle C is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the maximum and minimum values of (8x−x2−12​−4)2+(x−7)2,x∈R be M and m, respectively. Then M2−m2 is equal to _________

Center of Circle $C_{1}$

Radius of Circle $C_{1}$

Center of Circle $C_{2}$

Radius of Circle $C_{2}$

Consider $M$ with $r = \frac{( 2 ^{199} - 1 ) 2}{2 ^{198}}$ . The number of all those circles $D_{n}$ that are inside $M$ is

The set of values of $k$ for which the circle $C : 4 x^{2} + 4 y^{2} - 12 x + 8 y + k = 0$ lies inside the fourth quadrant and the point $(1, - \frac{1}{3})$ lies on or inside the circle $C$ is :

Let the maximum and minimum values of $(8 x - x^{2} - 12 - 4)^{2} + (x - 7)^{2}, x \in R$ be $M$ and $m$ , respectively. Then $M^{2} - m^{2}$ is equal to _________