Let the incentre of `DeltaABC` is I(2, 5). If `A=(1, 13)` and `B=(-4, 1)`, then the coordinates of C are

Options

A. `(1, 10)`
B. `(10, 1)`
C. `(8, 2)`
D. `(9, 2)`

Answer

To find the coordinates of point C in triangle ABC given the incenter I(2, 5), vertex A(1, 13), and vertex B(-4, 1), we can follow these steps:

### Step 1: Find the equation of line AB
The coordinates of points A and B are given as A(1, 13) and B(-4, 1).

To find the slope (m) of line AB:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 13}{-4 - 1} = \frac{-12}{-5} = \frac{12}{5} \]

Using point-slope form of the line equation \(y - y_1 = m(x - x_1)\), we can write the equation of line AB:
\[ y - 13 = \frac{12}{5}(x - 1) \]
Simplifying this gives:
\[ y = \frac{12}{5}x + \frac{53}{5} \]

### Step 2: Find the perpendicular distance from I(2, 5) to line AB
The formula for the distance \(d\) from a point \((x_0, y_0)\) to a line \(Ax + By + C = 0\) is:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]

First, we convert the line equation to the standard form \(Ax + By + C = 0\):
\[ \frac{12}{5}x - y + \frac{53}{5} = 0 \implies 12x - 5y + 53 = 0 \]
Here, \(A = 12\), \(B = -5\), and \(C = 53\).

Now, substituting \(I(2, 5)\):
\[ d = \frac{|12(2) - 5(5) + 53|}{\sqrt{12^2 + (-5)^2}} = \frac{|24 - 25 + 53|}{\sqrt{144 + 25}} = \frac{|52|}{13} = 4 \]

### Step 3: Find the coordinates of point C
Let the coordinates of C be \((x, y)\). The distance from I(2, 5) to line AC must also equal the inradius \(r = 4\).

Assuming the slope of line AC is \(m_1\), the equation of line AC can be written as:
\[ y - 13 = m_1(x - 1) \implies y = m_1x + 13 - m_1 \]

Using the distance formula again, we set the distance from I(2, 5) to line AC equal to 4:
\[ \frac{|m_1(2) - 5 + 13 - m_1|}{\sqrt{1 + m_1^2}} = 4 \]

This leads to:
\[ |(m_1 + 8)| = 4\sqrt{1 + m_1^2} \]

Squaring both sides and solving the resulting quadratic equation will give us the slope \(m_1\).

### Step 4: Solve for coordinates of C
Once we find \(m_1\), we can plug it back into the equation of line AC to find the coordinates of point C.

After going through the calculations, we find that the coordinates of C are:
\[ C(10, 1) \]

### Summary
The coordinates of point C are \((10, 1)\).