6.2 Q-4

Question Statement

Find the length of the tangent drawn from the point $(-5,4)$ to the circle given by the equation:

$$5x^2 + 5y^2 - 10x + 15y - 131 = 0$$

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Background and Explanation

To solve this problem, we need to recall the formula for the length of a tangent drawn from a point $(x_1, y_1)$ to a circle. The general equation of a circle is:

$$x^2 + y^2 + 2gx + 2fy + c = 0$$

The length of the tangent from a point $(x_1, y_1)$ to the circle is given by the formula:

$$\text{Length of Tangent} = \sqrt{x_1^2 + y_1^2 - 2gx_1 - 2fy_1 + c}$$

Where:

• $g$, $f$, and $c$ are the coefficients from the standard form of the circle equation.
• $(x_1, y_1)$ is the point outside the circle.

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Solution

We begin by simplifying the given equation of the circle:

$$5x^2 + 5y^2 - 10x + 15y - 131 = 0$$

Divide through by 5 to simplify:

$$x^2 + y^2 - 2x + 3y - \frac{131}{5} = 0$$

From this, we identify the coefficients:

• $g = -1$
• $f = \frac{3}{2}$
• $c = -\frac{131}{5}$

Now, using the formula for the length of the tangent:

$$\text{Length of Tangent} = \sqrt{(-5)^2 + 4^2 - 2(-5) + 3(4) - \frac{131}{5}}$$

We break this into parts:

1. $(-5)^2 = 25$
2. $4^2 = 16$
3. $-2(-5) = 10$
4. $3(4) = 12$

Substituting these values back into the equation:

$$\text{Length of Tangent} = \sqrt{25 + 16 + 10 + 12 - \frac{131}{5}}$$

Simplify further:

$$\text{Length of Tangent} = \sqrt{63 - \frac{131}{5}}$$

Now, to combine the terms under the square root:

$$63 = \frac{315}{5}$$

Thus:

$$\text{Length of Tangent} = \sqrt{\frac{315 - 131}{5}} = \sqrt{\frac{184}{5}}$$

This is the exact length of the tangent.

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Key Formulas or Methods Used

• Formula for Length of Tangent:

$$\text{Length of Tangent} = \sqrt{x_1^2 + y_1^2 - 2gx_1 - 2fy_1 + c}$$

• Standard form of a circle:

$$x^2 + y^2 + 2gx + 2fy + c = 0$$

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Summary of Steps

1. Write the given circle equation in the standard form by dividing through by 5.
2. Identify the values of $g$, $f$, and $c$.
3. Apply the formula for the length of the tangent using the given point $(-5, 4)$.
4. Simplify the expression step by step.
5. Final result:

$$\text{Length of Tangent} = \sqrt{\frac{184}{5}}$$