The diagonal of the rectangle formed by the lines `x^2-7x +6= 0` and `y^2-14y +40= 0` is

Options

A. `5x+6y=0`
B. `5x-6y=0`
C. `6x-5y+14=0`
D. `6x-5y-14=0`

Answer

To find the diagonal of the rectangle formed by the lines given by the equations \( x^2 - 7x + 6 = 0 \) and \( y^2 - 14y + 40 = 0 \), we will follow these steps:

### Step 1: Factor the equation \( x^2 - 7x + 6 = 0 \)

To factor this quadratic equation, we look for two numbers that multiply to \( 6 \) (the constant term) and add up to \( -7 \) (the coefficient of \( x \)).

\[ x^2 - 7x + 6 = (x - 1)(x - 6) = 0 \]

### Step 2: Solve for \( x \)

Setting each factor to zero gives us the solutions:

\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \]

### Step 3: Factor the equation \( y^2 - 14y + 40 = 0 \)

Similarly, we factor this quadratic equation by finding two numbers that multiply to \( 40 \) and add up to \( -14 \).

\[ y^2 - 14y + 40 = (y - 10)(y - 4) = 0 \]

### Step 4: Solve for \( y \)

Setting each factor to zero gives us the solutions:

\[ y - 10 = 0 \quad \Rightarrow \quad y = 10 \]
\[ y - 4 = 0 \quad \Rightarrow \quad y = 4 \]

### Step 5: Identify the vertices of the rectangle

The solutions give us the vertices of the rectangle formed by the lines:
- \( (1, 4) \)
- \( (1, 10) \)
- \( (6, 4) \)
- \( (6, 10) \)

### Step 6: Calculate the length and width of the rectangle

The length of the rectangle (along the x-axis) is:

\[ \text{Length} = 6 - 1 = 5 \]

The width of the rectangle (along the y-axis) is:

\[ \text{Width} = 10 - 4 = 6 \]

### Step 7: Calculate the diagonal using the Pythagorean theorem

The diagonal \( d \) of the rectangle can be calculated using the Pythagorean theorem:

\[ d = \sqrt{(\text{Length})^2 + (\text{Width})^2} \]
\[ d = \sqrt{5^2 + 6^2} = \sqrt{25 + 36} = \sqrt{61} \]

### Final Answer

The diagonal of the rectangle is:

\[ \sqrt{61} \]

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