Question:
The length of the tangent from the point $(3, 4)$ to the circle $x^2 + y^2 - 2x - 4y + 1 = 0$ is:
The length of the tangent from the point $(3, 4)$ to the circle $x^2 + y^2 - 2x - 4y + 1 = 0$ is:
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Length of tangent is always $\sqrt{S_{11}}$. Ensure the coefficients of $x^2$ and $y^2$ are $1$ before calculating the value of $S_{11}$.
Updated On: Jun 3, 2026
- $2$
- $4$
- $\sqrt{2}$
- $3$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
For a point $P(x_1, y_1)$ and a circle $S = x^2 + y^2 + 2gx + 2fy + c = 0$, the length of the tangent is given by $\sqrt{S_{11}}$, where: \[ S_{11} = x_1^2 + y_1^2 + 2g x_1 + 2f y_1 + c \]
Step 2: Meaning
We substitute the point $(3, 4)$ directly into the circle's equation to find $S_{11}$, and then calculate its square root.
Step 3: Analysis
Evaluate $S_{11}$ at $(3, 4)$: \[ S_{11} = 3^2 + 4^2 - 2(3) - 4(4) + 1 \] \[ S_{11} = 9 + 16 - 6 - 16 + 1 = 4 \] Length of the tangent: \[ L = \sqrt{S_{11}} = \sqrt{4} = 2 \]
Step 4: Conclusion
The length of the tangent from the point $(3, 4)$ to the circle is $2$ units.
Final Answer: (A)
For a point $P(x_1, y_1)$ and a circle $S = x^2 + y^2 + 2gx + 2fy + c = 0$, the length of the tangent is given by $\sqrt{S_{11}}$, where: \[ S_{11} = x_1^2 + y_1^2 + 2g x_1 + 2f y_1 + c \]
Step 2: Meaning
We substitute the point $(3, 4)$ directly into the circle's equation to find $S_{11}$, and then calculate its square root.
Step 3: Analysis
Evaluate $S_{11}$ at $(3, 4)$: \[ S_{11} = 3^2 + 4^2 - 2(3) - 4(4) + 1 \] \[ S_{11} = 9 + 16 - 6 - 16 + 1 = 4 \] Length of the tangent: \[ L = \sqrt{S_{11}} = \sqrt{4} = 2 \]
Step 4: Conclusion
The length of the tangent from the point $(3, 4)$ to the circle is $2$ units.
Final Answer: (A)
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