6.2 Q-9

Question Statement

Find the equation of the chord of contact of the tangents drawn from the point $(4, 5)$ to the circle given by:

$$2x^2 + 2y^2 - 8x + 12y + 21 = 0$$

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Background and Explanation

To solve this problem, we need to understand the concept of the chord of contact. The chord of contact is the line joining the points of tangency of two tangents drawn from an external point to a circle.

The equation of the chord of contact can be derived using the formula:

$$x x_1 + y y_1 + (g)(x + x_1) + (f)(y + y_1) + c = 0$$

Where:

• $(x_1, y_1)$ is the external point (in this case, $(4, 5)$),
• $(g, f, c)$ are the coefficients of the general equation of the circle $x^2 + y^2 + 2gx + 2fy + c = 0$.

We will also use the fact that the points of tangency $P(x_1, y_1)$ and $Q(x_2, y_2)$ satisfy the equation of the chord of contact.

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Solution

Step 1: Rewrite the Circle’s Equation in Standard Form

Given the circle’s equation:

$$2x^2 + 2y^2 - 8x + 12y + 21 = 0$$

Divide the entire equation by 2 to simplify:

$$x^2 + y^2 - 4x + 6y + \frac{21}{2} = 0 \tag{1}$$

Step 2: Derive the Equation of the Tangent at Any Point on the Circle

Let $P(x_1, y_1)$ and $Q(x_2, y_2)$ be the points of contact of the tangents. The equation of the tangent at any point on the circle is given by:

$$xx_1 + yy_1 + (-2)(x + x_1) + 3(y + y_1) + \frac{21}{2} = 0 \tag{2}$$

Step 3: Substitute the External Point $(4, 5)$ into the Tangent Equation

Substitute $x = 4$ and $y = 5$ into equation (2) to find the relationship between $x_1$ and $y_1$:

$$4x_1 + 5y_1 + (-2)(4 + x_1) + 3(5 + y_1) + \frac{21}{2} = 0$$

Simplifying the equation:

$$4x_1 + 5y_1 - 8 - 2x_1 + 15 + 3y_1 + \frac{21}{2} = 0$$ $$2x_1 + 8y_1 + 7 + \frac{21}{2} = 0$$

Multiply through by 2 to eliminate the fraction:

$$4x_1 + 16y_1 + 14 + 21 = 0$$

Simplify further:

$$4x_1 + 16y_1 + 35 = 0 \tag{3}$$

Step 4: Equation for the Second Point of Tangency $Q(x_2, y_2)$

By symmetry, the second point of tangency $Q(x_2, y_2)$ also satisfies a similar equation:

$$4x_2 + 16y_2 + 35 = 0 \tag{4}$$

Step 5: Conclusion - Equation of the Chord of Contact

Since both points $P(x_1, y_1)$ and $Q(x_2, y_2)$ lie on the line $4x + 16y + 35 = 0$, this line is the required equation of the chord of contact.

Thus, the equation of the chord of contact is:

$$4x + 16y + 35 = 0$$

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Key Formulas or Methods Used

• General Equation of the Tangent to a Circle:

$$xx_1 + yy_1 + (g)(x + x_1) + (f)(y + y_1) + c = 0$$

• Condition for Chord of Contact: The points of tangency lie on the line joining them, which is the chord of contact.

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Summary of Steps

1. Simplify the given equation of the circle to standard form.
2. Write the equation for the tangent at any point on the circle.
3. Substitute the coordinates of the external point $(4, 5)$ into the tangent equation.
4. Simplify the resulting equation to find the relationship between the coordinates of the points of tangency.
5. The equation of the chord of contact is $4x + 16y + 35 = 0$.