6.2 Q-5

Question Statement

Find the length of the chord cut off from the line $2x + 3y = 13$ by the circle $x^2 + y^2 = 26$.

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Background and Explanation

To solve this problem, we need to use both the equation of the line and the equation of the circle. The line cuts the circle at two points, creating a chord. To find the length of this chord, we first need to determine the points of intersection between the line and the circle. Once we have these points, we can use the distance formula to calculate the length of the chord.

Key concepts used:

• Equation of a line: $Ax + By + C = 0$
• Equation of a circle: $x^2 + y^2 = r^2$
• Distance between two points: Given points $(x_1, y_1)$ and $(x_2, y_2)$, the distance is calculated as:

$$l = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

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Solution

Step 1: Express $y$ in terms of $x$ using the line equation

Starting with the given line equation $2x + 3y = 13$, solve for $y$:

$$3y = 13 - 2x$$ $$y = \frac{13 - 2x}{3} \tag{1}$$

Step 2: Substitute the expression for $y$ into the circle equation

The equation of the circle is $x^2 + y^2 = 26$. Substitute the value of $y$ from equation (1) into the circle equation:

$$x^2 + \left( \frac{13 - 2x}{3} \right)^2 = 26$$

Step 3: Simplify the equation

Expand and simplify the terms:

$$x^2 + \frac{169 - 52x + 4x^2}{9} = 26$$

Multiply through by 9 to eliminate the denominator:

$$9x^2 + 169 - 52x + 4x^2 = 234$$

Combine like terms:

$$13x^2 - 52x - 65 = 0$$

Step 4: Solve the quadratic equation

To solve the quadratic equation $13x^2 - 52x - 65 = 0$, divide by 13:

$$x^2 - 4x - 5 = 0$$

Factor the quadratic:

$$(x - 5)(x + 1) = 0$$

Thus, $x = 5$ or $x = -1$.

Step 5: Find the corresponding $y$-values

Substitute these $x$-values into equation (1) to find the corresponding $y$-values:

For $x = -1$:

$$y = \frac{13 + 2}{3} = \frac{15}{3} = 5$$

For $x = 5$:

$$y = \frac{13 - 10}{3} = \frac{3}{3} = 1$$

So, the points of intersection are $(5, 1)$ and $(-1, 5)$.

Step 6: Calculate the length of the chord

Now that we have the points of intersection, we can use the distance formula to find the length of the chord. The points are $(5, 1)$ and $(-1, 5)$, so the length of the chord is:

$$l = \sqrt{(5 - (-1))^2 + (1 - 5)^2}$$

Simplify:

$$l = \sqrt{(6)^2 + (-4)^2} = \sqrt{36 + 16} = \sqrt{52}$$

Finally, express the result:

$$l = 2 \sqrt{13}$$

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Key Formulas or Methods Used

• Distance Formula:

$$l = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

• Equation of a Circle:

$$x^2 + y^2 = r^2$$

• Solving Quadratic Equations: Factorization method for solving quadratic equations.

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Summary of Steps

1. Rearrange the line equation to express $y$ in terms of $x$.
2. Substitute the expression for $y$ into the circle equation.
3. Simplify the resulting equation and solve the quadratic.
4. Find the $y$-coordinates for the points of intersection.
5. Use the distance formula to calculate the length of the chord:

$$l = 2 \sqrt{13}$$