JEE Advanced 2008

LEVELJEE Main

Animated Solution for Mathematics - Circles: Consider $L_{1} : 2 x + 3 y + p - 3 = 0$ , $L_{2} : 2 x + 3 y + p + 3 = 0$ where $p$ is a real number, and $C : x^{2} + y^{2} + 6 x - 10 y + 30 = 0$ . STATEMENT-1 : If line $L_{1}$ is a chord of circle $C$ , then line $L_{2}$ is not always a diameter of circle $C$ and STATEMENT-2 : If line $L_{1}$ is a diameter of circle $C$ , then line $L_{2}$ is not a chord of circle $C$ .

Q: Wait, why is the distance between the lines independent of $p$?

Great question! The distance between two parallel lines $ax + by + c_1 = 0$ and $ax + by + c_2 = 0$ is given by $\frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}}$. When you subtract the constant terms $(p+3) - (p-3)$, the variable $p$ cancels out completely. This means the lines are always a fixed distance apart, no matter how you shift them by changing $p$.

Q: How do I know if a line is a chord or a diameter?

Think of the center of the circle as the origin of a target. A diameter is a line that passes exactly through the center, so its perpendicular distance from the center is $0$. A chord is any line that cuts through the circle, meaning its perpendicular distance from the center must be less than the radius. If the distance is equal to the radius, it is a tangent. If it is greater, it doesn't touch the circle at all.

Q: If $L_1$ is a diameter, why must $L_2$ be a chord?

If $L_1$ is a diameter, it passes through the center, so its distance from the center is $0$. Since $L_2$ is parallel to $L_1$ and they are separated by a fixed distance $d \approx 1.66$, $L_2$ must be at a distance of $1.66$ from the center. Because the radius of our circle is $2$, and $1.66 < 2$, $L_2$ must pass through the interior of the circle, making it a chord by definition.

Select Answer:

Visualized Solution

Identify the Circle's Parameters

We are given the circle equation: $C : x^{2} + y^{2} + 6 x - 10 y + 30 = 0$
Compare this with the standard general equation: $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$
Here, $2 g = 6 ⟹ g = 3$ and $2 f = - 10 ⟹ f = - 5$
The constant term is $c = 30$

Calculate Center and Radius of $C$

Center of the circle: $(- g, - f) = (- 3, 5)$
Radius formula: $r = g^{2} + f^{2} - c$
Substitute values: $r = 3^{2} + (- 5)^{2} - 30$
Simplify: $r = 9 + 25 - 30 = 4 = 2$

Analyze the Equations of $L_{1}$ and $L_{2}$

Line $L_{1} : 2 x + 3 y + p - 3 = 0$
Line $L_{2} : 2 x + 3 y + p + 3 = 0$
Observe the coefficients of $x$ and $y$ are identical: $2$ and $3$
This means both lines have the same slope: $m = - \frac{2}{3}$
Therefore, $L_{1}$ and $L_{2}$ are parallel lines ( $L_{1} ∥ L_{2}$ )

Formula for Distance Between Parallel Lines

For two parallel lines $a x + b y + c_{1} = 0$ and $a x + b y + c_{2} = 0$ :
The perpendicular distance $d$ is given by: $d = \frac{∣ c _{1} - c _{2} ∣}{a ^{2} + b ^{2}}$
Here, $a = 2$ , $b = 3$ , $c_{1} = p - 3$ , and $c_{2} = p + 3$

Compute the Perpendicular Distance $d$

Substitute the constants into the formula:
$d = \frac{∣ ( p + 3 ) - ( p - 3 ) ∣}{2 ^{2} + 3 ^{2}}$
Simplify the numerator: $∣ p + 3 - p + 3∣ = 6$
Simplify the denominator: $4 + 9 = 13$
Therefore, $d = \frac{6}{13} \approx 1.66$

Analyze Statement-2: $L_{1}$ as a Diameter

Statement-2 states: "If line $L_{1}$ is a diameter of circle $C$ , then line $L_{2}$ is not a chord of circle $C$ ."
If $L_{1}$ is a diameter, it must pass through the center $(- 3, 5)$ .
The distance from the center to $L_{1}$ is $0$ .
Since $L_{2}$ is parallel to $L_{1}$ at a distance of $d = \frac{6}{13}$ , the distance of $L_{2}$ from the center is also exactly $d \approx 1.66$ .

Compare Distance with Radius for $L_{2}$

Distance of $L_{2}$ from center: $d \approx 1.66$
Radius of the circle: $r = 2$
Since $d < r$ (i.e., $1.66 < 2$ ), the line $L_{2}$ lies inside the circle.
Any line whose distance from the center is less than the radius is a chord.
Therefore, $L_{2}$ must be a chord.
This directly contradicts Statement-2, so Statement-2 is False.

Analyze Statement-1: $L_{1}$ as a Chord

Statement-1 states: "If line $L_{1}$ is a chord of circle $C$ , then line $L_{2}$ is not always a diameter of circle $C$ ."
If $L_{1}$ is a chord, its distance $h$ from the center is less than the radius: $h < 2$ .
For $L_{2}$ to be a diameter, its distance from the center must be exactly $0$ .
This can only happen if the distance between the lines, $d = 1.66$ , is exactly equal to the distance of $L_{1}$ from the center, $h$ .

Final Verdict on Statement-1 and Option Selection

Since $L_{1}$ can be any chord (with distance $h \in [0, 2)$ ), $L_{2}$ will only be a diameter when $h = \frac{6}{13}$ .
For all other positions of the chord $L_{1}$ , $L_{2}$ will not be a diameter.
Thus, $L_{2}$ is indeed not always a diameter.
Therefore, Statement-1 is True.
Correct Option: Statement-1 is True, Statement-2 is False

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

Solution Diagram

Analyzing the Circle

To understand the geometry of the circle

C

, we begin with the given equation:

x^{2} + y^{2} + 6 x - 10 y + 30 = 0

By comparing this to the standard form

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

, we identify the parameters

g = 3

and

f = - 5

. The center of the circle is located at

(- g, - f) = (- 3, 5)

The radius

r

is calculated as follows:

r = g^{2} + f^{2} - c = 3^{2} + (- 5)^{2} - 30 = 9 + 25 - 30 = 4 = 2

The Geometry of Parallel Lines

We are given two lines,

L_{1} : 2 x + 3 y + p - 3 = 0

and

L_{2} : 2 x + 3 y + p + 3 = 0

. Because the coefficients of

x

and

y

are identical, these lines are parallel with a slope of

m = - \frac{2}{3}

The perpendicular distance

d

between these two parallel lines is given by the formula:

d = \frac{∣ c _{1} - c _{2} ∣}{a ^{2} + b ^{2}}

Substituting the given values

a = 2

b = 3

c_{1} = p - 3

, and

c_{2} = p + 3

, we find:

d = \frac{∣ ( p + 3 ) - ( p - 3 ) ∣}{2 ^{2} + 3 ^{2}} = \frac{6}{13}

Since

13 \approx 3.6

, the distance

d \approx 1.66

. This represents the constant gap between the two lines.

Evaluating the Statements

Statement-2 Analysis: If

L_{1}

is a diameter, it must pass through the center

(- 3, 5)

, meaning its perpendicular distance from the center is

0

. Since

L_{2}

is parallel to

L_{1}

at a distance of

d \approx 1.66

, the distance of

L_{2}

from the center is exactly

1.66

Because

1.66 < 2

(the radius of the circle),

L_{2}

must lie inside the circle and intersect it at two points. Therefore,

L_{2}

is necessarily a chord, which makes Statement-2 False.

Statement-1 Analysis: If

L_{1}

is a chord, its distance from the center

h

can be any value in the range

[0, 2)

. For

L_{2}

to be a diameter, its distance from the center must be

0

This condition only occurs if

L_{1}

is at a distance of

d = 1.66

from the center. Since

L_{1}

can be any chord within the circle,

L_{2}

is not always a diameter. Therefore, Statement-1 is True.

Final Conclusion

By evaluating the geometric constraints, we conclude that Statement-1 is True, and Statement-2 is False.

Common Traps & FAQs

Wait, why is the distance between the lines independent of

p

Great question! The distance between two parallel lines

a x + b y + c_{1} = 0

and

a x + b y + c_{2} = 0

is given by

\frac{∣ c _{1} - c _{2} ∣}{a ^{2} + b ^{2}}

. When you subtract the constant terms

(p + 3) - (p - 3)

, the variable

p

cancels out completely. This means the lines are always a fixed distance apart, no matter how you shift them by changing

p

How do I know if a line is a chord or a diameter?

Think of the center of the circle as the origin of a target. A diameter is a line that passes exactly through the center, so its perpendicular distance from the center is

0

. A chord is any line that cuts through the circle, meaning its perpendicular distance from the center must be less than the radius. If the distance is equal to the radius, it is a tangent. If it is greater, it doesn't touch the circle at all.

L_{1}

is a diameter, why must

L_{2}

be a chord?

L_{1}

is a diameter, it passes through the center, so its distance from the center is

0

. Since

L_{2}

is parallel to

L_{1}

and they are separated by a fixed distance

d \approx 1.66

L_{2}

must be at a distance of

1.66

from the center. Because the radius of our circle is

2

, and

1.66 < 2

L_{2}

must pass through the interior of the circle, making it a chord by definition.

Select Answer:

Visualized Solution

Identify the Circle's Parameters

Calculate Center and Radius of $C$

Analyze the Equations of $L_{1}$ and $L_{2}$

Formula for Distance Between Parallel Lines

Compute the Perpendicular Distance $d$

Analyze Statement-2: $L_{1}$ as a Diameter

Compare Distance with Radius for $L_{2}$

Analyze Statement-1: $L_{1}$ as a Chord

Final Verdict on Statement-1 and Option Selection

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

Analyzing the Circle

The Geometry of Parallel Lines

Evaluating the Statements

Final Conclusion

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Comprehension Passage

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Select Answer:

Visualized Solution

Identify the Circle's Parameters

Calculate Center and Radius of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C

Analyze the Equations of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } L1​ and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } L2​

Formula for Distance Between Parallel Lines

Compute the Perpendicular Distance .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } d

Analyze Statement-2: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } L1​ as a Diameter

Compare Distance with Radius for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } L2​

Analyze Statement-1: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } L1​ as a Chord

Final Verdict on Statement-1 and Option Selection

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

Analyzing the Circle

The Geometry of Parallel Lines

Evaluating the Statements

Final Conclusion

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Calculate Center and Radius of $C$

Analyze the Equations of $L_{1}$ and $L_{2}$

Compute the Perpendicular Distance $d$

Analyze Statement-2: $L_{1}$ as a Diameter

Compare Distance with Radius for $L_{2}$

Analyze Statement-1: $L_{1}$ as a Chord

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