Two lines through `(2, 3)` from which the circle `x^2+y^2 =25` intercepts chords of length `8` units have equations
(A) `2x+3y=13`, `x+5y=17`
(B) `y= 3`, `12x+5y=39`
(C) `x=2`, `9x -11y=51`
(D) `y=0`, `12x+5y=39`

Options

A. `2x+3y=13, x+5y=17`
B. `y=3, 12x+5y=39`
C. `x=2, 9x-11y=51`
D. y
=
0
,
12
x
+
5
y
=
39

Answer

To solve the problem, we need to find the equations of two lines that pass through the point (2, 3) and intercept chords of length 8 units in the circle defined by the equation \(x^2 + y^2 = 25\).

### Step 1: Understand the Circle
The given circle has the equation \(x^2 + y^2 = 25\). This implies that the radius of the circle is \(5\) (since \(r^2 = 25\)).

**Hint:** The radius of a circle is the square root of the constant term in the circle's equation.

### Step 2: Length of Chord Formula
The length of the chord intercepted by a line \(y = mx + c\) in a circle \(x^2 + y^2 = r^2\) is given by:
\[ L = \frac{2\sqrt{r^2 - c^2}}{\sqrt{1 + m^2}} \]
where \(L\) is the length of the chord, \(r\) is the radius, and \(c\) is the y-intercept of the line.

**Hint:** Remember to substitute \(r = 5\) into the formula.

### Step 3: Set Up the Equation
In our case, the length of the chord \(L\) is given as \(8\). Therefore, we set up the equation:
\[ 8 = \frac{2\sqrt{25 - c^2}}{\sqrt{1 + m^2}} \]

**Hint:** Rearranging the equation will help isolate terms involving \(c\) and \(m\).

### Step 4: Simplify the Equation
Squaring both sides gives:
\[ 64(1 + m^2) = 4(25 - c^2) \]
This simplifies to:
\[ 64 + 64m^2 = 100 - 4c^2 \]
Rearranging gives:
\[ 4c^2 + 64m^2 = 36 \]
Dividing through by 4:
\[ c^2 + 16m^2 = 9 \]

**Hint:** This is a standard form of an equation that relates \(c\) and \(m\).

### Step 5: Use the Point (2, 3)
Since the line passes through the point (2, 3), we can express \(c\) in terms of \(m\):
\[ c = 3 - 2m \]

**Hint:** Substitute this expression for \(c\) back into the equation \(c^2 + 16m^2 = 9\).

### Step 6: Substitute and Solve for \(m\)
Substituting \(c = 3 - 2m\) into the equation:
\[ (3 - 2m)^2 + 16m^2 = 9 \]
Expanding gives:
\[ 9 - 12m + 4m^2 + 16m^2 = 9 \]
Combining like terms:
\[ 20m^2 - 12m = 0 \]
Factoring out \(4m\):
\[ 4m(5m - 3) = 0 \]
Thus, \(m = 0\) or \(m = \frac{3}{5}\).

**Hint:** Finding the values of \(m\) helps us determine the slopes of the lines.

### Step 7: Find Corresponding \(c\) Values
For \(m = 0\):
\[ c = 3 - 2(0) = 3 \quad \Rightarrow \quad y = 3 \]

For \(m = \frac{3}{5}\):
\[ c = 3 - 2\left(\frac{3}{5}\right) = 3 - \frac{6}{5} = \frac{9}{5} \quad \Rightarrow \quad 5y = 12x + 39 \quad \Rightarrow \quad 12x + 5y = 39 \]

**Hint:** Ensure that the equations are in the correct form.

### Final Result
The two lines are:
1. \(y = 3\)
2. \(12x + 5y = 39\)

Thus, the correct option is **(B)**.