A Question is given followed by two Statements I and II. Consider the Question and the Statements. Question: AB and CD are chords of a circle intersecting at P. If $AP \times PB$ = 48 square units, then what is $CP \times PD$ equal to? Statement-I : AP = 8 units Statement-II: CP = 10 units Which one of the following is correct in respect of the above Question and the Statements?

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Question

CDS - 2025

A Question is given followed by two Statements I and II. Consider the Question and the Statements.
Question:
AB and CD are chords of a circle intersecting at P. If $AP \times PB$ = 48 square units, then what is $CP \times PD$ equal to?
Statement-I : AP = 8 units
Statement-II: CP = 10 units
Which one of the following is correct in respect of the above Question and the Statements?

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CDS 2 2025 Maths Question Paper (14-Sep-2025)

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The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone

The Question can be answered by using either Statement alone

The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone

The Question can be answered even without using any of the Statements

The correct answer is

The Question can be answered even without using any of the Statements

Circle Geometry: Intersecting Chords Theorem

This question requires understanding a specific property of chords intersecting inside a circle. We are given the product of the segments of one chord and need to find the product of the segments of the other intersecting chord.

Intersecting Chords Theorem Explained

The core concept needed here is the Intersecting Chords Theorem. This theorem is a fundamental result in Euclidean geometry concerning circles. It states that if two chords of a circle intersect internally, then the product of the lengths of the segments on each chord are equal.

Mathematically, if two chords AB and CD intersect at a point P inside a circle, the theorem is expressed as:

$AP \\times PB = CP \\times PD$

This equality holds true regardless of the specific lengths of the segments, as long as they form intersecting chords within the same circle.

Applying the Theorem

The question provides the following information:

AB and CD are chords of a circle intersecting at P.
The product $AP \\times PB = 48$ square units.
We need to determine the value of $CP \\times PD$.

Using the Intersecting Chords Theorem, we can directly equate the products of the segments:

$CP \\times PD = AP \\times PB$

Given that $AP \\times PB = 48$, we substitute this value into the equation:

$CP \\times PD = 48$

Therefore, the value of $CP \\times PD$ is 48 square units. This answer is derived directly from the information provided in the question statement itself.

Evaluating the Statements

Now, let's assess the role of the two statements provided:

Statement I: AP = 8 units.
Statement II: CP = 10 units.

Statement I gives the length of one segment of chord AB. If $AP = 8$, then $8 \\times PB = 48$, which means $PB = 6$. While this tells us the specific lengths of the segments of chord AB, it is not needed to find the value of $CP \\times PD$, which we already established is 48.

Statement II gives the length of one segment of chord CD. If $CP = 10$, then $10 \\times PD = 48$, meaning $PD = 4.8$. Similar to Statement I, this information provides specific segment lengths for chord CD but is redundant for determining the product $CP \\times PD$.

Since the value of $CP \\times PD$ can be determined solely from the initial premise ($AP \\times PB = 48$) using the Intersecting Chords Theorem, neither Statement I nor Statement II is necessary.

Final Conclusion on Sufficiency

The question asks for the value of $CP \\times PD$. The Intersecting Chords Theorem provides a direct relationship: $CP \\times PD = AP \\times PB$. The value of $AP \\times PB$ is given as 48 in the question itself. Therefore, the question is answerable using only the information presented in the question, without recourse to either Statement I or Statement II.

This corresponds to the conclusion that the question can be answered even without using any of the statements.

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