The angle between the chords of the circle `x^2 + y^2 = 100`, which passes through the point (7,1) and also divides the circumference of the circle into two arcs whose length are in the ratio 2 : 1, is equal to

Options

A. `pi/6`
B. `pi/3`
C. `pi/2`
D. `(2pi)/3`

Answer

To solve the problem, we need to find the angle between the chords of the circle \(x^2 + y^2 = 100\) that pass through the point (7, 1) and divide the circumference into two arcs in the ratio of 2:1.

### Step-by-Step Solution:

1. **Identify the Circle and its Properties**:
The equation of the circle is given as \(x^2 + y^2 = 100\). This represents a circle with center at the origin (0, 0) and a radius \(r = 10\) (since \(\sqrt{100} = 10\)).

**Hint**: Remember that the radius can be found by taking the square root of the constant term in the circle's equation.

2. **Determine the Angle Subtended by the Chords**:
The problem states that the chords divide the circumference into two arcs in the ratio of 2:1. This means that the angle subtended by the chord at the center of the circle is \(120^\circ\) (since \(360^\circ \times \frac{2}{3} = 240^\circ\) for one arc and \(360^\circ \times \frac{1}{3} = 120^\circ\) for the other arc).

**Hint**: Use the property that the total angle around a point is \(360^\circ\) to find the angles corresponding to the arcs.

3. **Calculate the Height from the Center to the Chord**:
Let \(h\) be the perpendicular distance from the center (0, 0) to the chord. We can use the cosine of the angle subtended at the center to find this distance. For \(60^\circ\) (half of \(120^\circ\)), we have:
\[ \cos(60^\circ) = \frac{h}{10} \]
Since \(\cos(60^\circ) = \frac{1}{2}\), we can set up the equation:
\[ \frac{1}{2} = \frac{h}{10} \implies h = 5 \]

**Hint**: Use trigonometric ratios to relate the angle subtended at the center to the height from the center to the chord.

4. **Equation of the Chord**:
The chord can be expressed in point-slope form. The point (7, 1) lies on the chord, and we can express the equation of the chord as:
\[ y - 1 = m(x - 7) \]
Rearranging gives us:
\[ mx - y + (1 - 7m) = 0 \]

**Hint**: Remember that the slope-intercept form can be transformed into standard form for easier manipulation.

5. **Distance from the Center to the Chord**:
The distance \(d\) from the center (0, 0) to the chord can be calculated using the formula:
\[ d = \frac{|c|}{\sqrt{a^2 + b^2}} \]
where \(ax + by + c = 0\) is the equation of the line. Here, \(a = m\), \(b = -1\), and \(c = 1 - 7m\). Setting this distance equal to \(5\):
\[ 5 = \frac{|1 - 7m|}{\sqrt{m^2 + 1}} \]

**Hint**: Use the distance formula for a point to a line to relate the distance from the center to the chord.

6. **Solving for the Slope \(m\)**:
Squaring both sides and simplifying leads to a quadratic equation in \(m\):
\[ (1 - 7m)^2 = 25(m^2 + 1) \]
Expanding and rearranging gives:
\[ 24m^2 - 14m - 24 = 0 \]

**Hint**: When dealing with quadratic equations, remember to use the quadratic formula if necessary.

7. **Finding the Angle Between the Chords**:
The slopes of the two chords can be found from the roots of the quadratic. The angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) can be found using:
\[ \tan(\theta) = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right| \]
Since the product of the slopes is \(-1\) (indicating perpendicular lines), the angle between the chords is \(90^\circ\) or \(\frac{\pi}{2}\) radians.

**Hint**: Use the relationship between slopes and angles to find the angle between the two lines.

### Final Answer:
The angle between the chords is \(\frac{\pi}{2}\) radians.