Question Statement

Find the length of the tangent to the circle $x^2+y^2-18x+16y+10=0$ from the point $(-1,-1)$ lying outside the circle.

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Background and Explanation

The length of a tangent from an external point $(x_1,y_1)$ to a circle $x^2+y^2+Dx+Ey+F=0$ is given by $\sqrt{x_1^2+y_1^2+D{x}_1+E{y}_1+F}$. This formula is derived from the Pythagorean theorem, where the tangent forms a right angle with the radius at the point of contact, creating a right triangle with the distance from the center to the external point.

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Solution

We are given the circle equation $x^2+y^2-18x+16y+10=0$ and the external point $\left(x_1,y_1\right)=(-1,-1)$.

The length of the tangent from an external point to a circle is found by substituting the point coordinates into the left-hand side of the circle equation and taking the square root. Applying this formula:

$\begin{array}{rl}\text{Length}& =\sqrt{x_1^2+y_1^2-18x_1+16y_1+10}\\ & =\sqrt{(-1{)}^2+(-1{)}^2-18(-1)+16(-1)+10}\\ & =\sqrt{1+1+18-16+10}\\ & =\sqrt{14}\text{ units }\end{array}$

Therefore, the length of the tangent from the point $(-1,-1)$ to the given circle is $\sqrt{14}$ units.

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Key Formulas or Methods Used

• Length of tangent from external point: For a circle $x^2+y^2+Dx+Ey+F=0$ and point $(x_1,y_1)$ outside the circle, the tangent length is: $L=\sqrt{x_1^2+y_1^2+D{x}_1+E{y}_1+F}$
• Evaluation of algebraic expressions: Substituting values and simplifying arithmetic operations

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Summary of Steps

1. Identify the circle equation $x^2+y^2-18x+16y+10=0$ and the external point $(-1,-1)$.
2. Apply the tangent length formula by substituting the point into the circle expression.
3. Calculate: $(-1{)}^2+(-1{)}^2-18(-1)+16(-1)+10=1+1+18-16+10$.
4. Simplify the sum to get $14$ inside the radical.
5. Take the square root to obtain the final answer: $\sqrt{14}$ units.