Circles

Hardmathematics

The length of tangent from point (5, 4) to circle x² + y² = 9 is

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√32

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Subject: mathematics
Chapter: coordinate geometry
Topic: circles
Difficulty: Hard
Year: 2025

Solution

Correct Answer:

√32

Formula: The length of a tangent $L$ from an external point $(x_{1}, y_{1})$ to circle $x^{2} + y^{2} - r^{2} = 0$ is $L = x_{1}^{2} + y_{1}^{2} - r^{2}$ .
Identify Data: Point is $(5, 4)$ , Circle is $x^{2} + y^{2} - 9 = 0$ ( $r^{2} = 9$ ).
Calculate Power of Point:
- $S_{1} = 5^{2} + 4^{2} - 9 = 25 + 16 - 9 = 32$ .
Extract Length:
- $L = 32 = 42$ .

This hard difficulty mathematics question is from the chapter coordinate geometry, covering the topic of circles. It appeared in the 2025 exam.

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