Question Statement

Find the length of the tangent to the circle $3x^2+3y^2+18x-24y+50=0$ drawn from the point $(-3,1)$ lying outside the circle.

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Background and Explanation

To find the length of a tangent from an external point $(x_1,y_1)$ to a circle, the equation of the circle must first be in its general form $x^2+y^2+2gx+2fy+c=0$, where the coefficients of $x^2$ and $y^2$ are equal to $1$. The length of the tangent is then given by the square root of the power of the point with respect to the circle, denoted as $\sqrt{S_1}$.

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Solution

The given equation of the circle is: $3x^2+3y^2+18x-24y+50=0$

The given external point is $(x_1,y_1)=(-3,1)$.

Step 1: Normalize the Circle Equation

Before applying the formula for the length of the tangent, we must ensure the coefficients of $x^2$ and $y^2$ are $1$. We do this by dividing the entire equation by $3$: $\frac{3x^2}{3}+\frac{3y^2}{3}+\frac{18x}{3}-\frac{24y}{3}+\frac{50}{3}=0$ $x^2+y^2+6x-8y+\frac{50}{3}=0$

Step 2: Apply the Tangent Length Formula

The formula for the length of the tangent from a point $(x_1,y_1)$ to the circle $x^2+y^2+2gx+2fy+c=0$ is: $\text{Length}=\sqrt{x_1^2+y_1^2+2g{x}_1+2f{y}_1+c}$

Substituting the coordinates of the point $(-3,1)$ into the normalized equation: $\text{Length}=\sqrt{(-3{)}^2+(1{)}^2+6(-3)-8(1)+\frac{50}{3}}$

Step 3: Simplify the Expression

Now, we perform the arithmetic operations inside the square root: $\text{Length}=\sqrt{9+1-18-8+\frac{50}{3}}$ $\text{Length}=\sqrt{10-26+\frac{50}{3}}$ $\text{Length}=\sqrt{-16+\frac{50}{3}}$

To combine the terms, find a common denominator: $\text{Length}=\sqrt{\frac{-16\times 3+50}{3}}$ $\text{Length}=\sqrt{\frac{-48+50}{3}}$ $\text{Length}=\sqrt{\frac{2}{3}}\text{ units}$

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Key Formulas or Methods Used

• General Equation of a Circle: $x^2+y^2+2gx+2fy+c=0$
• Length of Tangent Formula: $L=\sqrt{S_1}=\sqrt{x_1^2+y_1^2+2g{x}_1+2f{y}_1+c}$
• Normalization: Dividing the equation by the coefficient of $x^2$ (if it is not 1) before calculation.

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Summary of Steps

1. Normalize the equation: Divide the circle equation by $3$ so that $x^2$ and $y^2$ have coefficients of $1$.
2. Identify the point: Use the given coordinates $(-3,1)$ as $(x_1,y_1)$.
3. Substitute into the formula: Plug the point into the expression $\sqrt{x^2+y^2+6x-8y+\frac{50}{3}}$.
4. Calculate the result: Simplify the numerical values to find the final length $\sqrt{\frac{2}{3}}$ units.