Question

The length of the tangent from the point $(3, 4)$ to the circle $x^2 + y^2 - 2x - 4y + 1 = 0$ is:

• A. $2$
• B. $4$
• C. $\sqrt{2}$
• D. $3$

Hint:

Length of tangent is always $\sqrt{S_{11}}$. Ensure the coefficients of $x^2$ and $y^2$ are $1$ before calculating the value of $S_{11}$.

Answer 1

The Correct Option is A

Step 1: Recall the tangent length rule.
The length of the tangent from a point $(x_1,y_1)$ to a circle $S = 0$ is the square root of the value you get by putting the point into the circle expression. We call that value $S_{11}$.

Step 2: Write the circle expression.
The circle is $x^2 + y^2 - 2x - 4y + 1 = 0$. To get $S_{11}$, we just put $(3, 4)$ into the left side.

Step 3: Substitute the point.
Replace $x$ with $3$ and $y$ with $4$.
\[ S_{11} = 3^2 + 4^2 - 2(3) - 4(4) + 1 \]

Step 4: Work out each term.
Compute carefully: $9 + 16 - 6 - 16 + 1$.

Step 5: Add them up.
Adding gives $9 + 16 = 25$, then $25 - 6 = 19$, then $19 - 16 = 3$, then $3 + 1 = 4$. So $S_{11} = 4$.

Step 6: Take the square root.
The tangent length is $\sqrt{S_{11}} = \sqrt{4} = 2$.
\[ \boxed{2} \]