Consider with circle `S: x^2+y^2-4x-1=0` and the line `L: y=3x-1`. If the line L cuts the circle at A and B then Length of the chord AB is

Options

A. `sqrt5`
B. `sqrt10`
C. `2sqrt5`
D. `5sqrt2`

Answer

To find the length of the chord AB where the line \( L: y = 3x - 1 \) intersects the circle \( S: x^2 + y^2 - 4x - 1 = 0 \), we can follow these steps:

### Step 1: Rewrite the Circle Equation
First, we will rewrite the equation of the circle in standard form. The given equation is:

\[ x^2 + y^2 - 4x - 1 = 0 \]

We can complete the square for the \( x \) terms:

\[ (x^2 - 4x) + y^2 = 1 \]

Completing the square for \( x^2 - 4x \):

\[ (x - 2)^2 - 4 + y^2 = 1 \]

This simplifies to:

\[ (x - 2)^2 + y^2 = 5 \]

This tells us that the center of the circle is \( (2, 0) \) and the radius \( r \) is \( \sqrt{5} \).

### Step 2: Find the Perpendicular Distance from the Center to the Line
Next, we need to find the perpendicular distance from the center of the circle \( (2, 0) \) to the line \( L: y = 3x - 1 \). We can rewrite the line in standard form:

\[ 3x - y - 1 = 0 \]

Using the formula for the distance \( d \) from a point \( (x_0, y_0) \) to a line \( Ax + By + C = 0 \):

\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]

Here, \( A = 3 \), \( B = -1 \), \( C = -1 \), and the point is \( (2, 0) \):

\[ d = \frac{|3(2) - 1(0) - 1|}{\sqrt{3^2 + (-1)^2}} = \frac{|6 - 1|}{\sqrt{9 + 1}} = \frac{5}{\sqrt{10}} = \frac{5\sqrt{10}}{10} = \frac{\sqrt{10}}{2} \]

### Step 3: Use Pythagorean Theorem to Find Length of Chord
Let \( l \) be half the length of the chord AB. According to the Pythagorean theorem:

\[ r^2 = d^2 + l^2 \]

Substituting the known values:

\[ (\sqrt{5})^2 = \left(\frac{\sqrt{10}}{2}\right)^2 + l^2 \]

This simplifies to:

\[ 5 = \frac{10}{4} + l^2 \]

\[ 5 = \frac{5}{2} + l^2 \]

Subtract \( \frac{5}{2} \) from both sides:

\[ 5 - \frac{5}{2} = l^2 \]

Convert \( 5 \) to a fraction:

\[ \frac{10}{2} - \frac{5}{2} = l^2 \]

\[ \frac{5}{2} = l^2 \]

Taking the square root:

\[ l = \sqrt{\frac{5}{2}} = \frac{\sqrt{10}}{2} \]

### Step 4: Find the Length of Chord AB
The total length of the chord AB is:

\[ AB = 2l = 2 \times \frac{\sqrt{10}}{2} = \sqrt{10} \]

### Final Answer
The length of the chord AB is \( \sqrt{10} \).

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