Find the bisector of the acute angle between the lines :
`5x=12y+24 and 12x=5y+10`

Answer

To find the bisector of the acute angle between the lines given by the equations \(5x = 12y + 24\) and \(12x = 5y + 10\), we can follow these steps:

### Step 1: Rewrite the equations in standard form
We start by rewriting both equations in the standard form \(Ax + By + C = 0\).

1. For the first line:
\[ 5x - 12y - 24 = 0 \]
Here, \(A_1 = 5\), \(B_1 = -12\), and \(C_1 = -24\).

2. For the second line:
\[ 12x - 5y - 10 = 0 \]
Here, \(A_2 = 12\), \(B_2 = -5\), and \(C_2 = -10\).

### Step 2: Calculate \(A_1 A_2 + B_1 B_2\)
Next, we need to calculate \(A_1 A_2 + B_1 B_2\) to determine the sign for the acute angle bisector.

\[ A_1 A_2 + B_1 B_2 = (5)(12) + (-12)(-5) = 60 + 60 = 120 \]

Since \(A_1 A_2 + B_1 B_2 > 0\), we will use the negative sign for the acute angle bisector.

### Step 3: Write the equation of the angle bisector
The equation of the angle bisector can be expressed as:
\[ \frac{A_1 x + B_1 y + C_1}{\sqrt{A_1^2 + B_1^2}} = \pm \frac{A_2 x + B_2 y + C_2}{\sqrt{A_2^2 + B_2^2}} \]

Substituting the values we have:
\[ \frac{5x - 12y - 24}{\sqrt{5^2 + (-12)^2}} = -\frac{12x - 5y - 10}{\sqrt{12^2 + (-5)^2}} \]

Calculating the denominators:
\[ \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]
\[ \sqrt{12^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]

### Step 4: Substitute the values into the bisector equation
Now substituting back into the equation:
\[ \frac{5x - 12y - 24}{13} = -\frac{12x - 5y - 10}{13} \]

Multiplying both sides by 13 to eliminate the denominator:
\[ 5x - 12y - 24 = - (12x - 5y - 10) \]

### Step 5: Simplify the equation
Expanding the right side:
\[ 5x - 12y - 24 = -12x + 5y + 10 \]

Bringing all terms to one side:
\[ 5x + 12x - 12y - 5y - 24 - 10 = 0 \]
\[ 17x - 17y - 34 = 0 \]

### Step 6: Finalize the equation
Dividing the entire equation by 17:
\[ x - y - 2 = 0 \]

Thus, the equation of the bisector of the acute angle between the lines is:
\[ \boxed{x - y - 2 = 0} \]