Question

Let A be the point (1, 2) and B be any point on the curve \( x^2 + y^2 = 16 \). If the centre of the locus of the point P, which divides the line segment AB in the ratio 3:2 is the point C (\( \alpha, \beta \)), then the length of the line segment AC is:

• A. \( \frac{6 \sqrt{5}}{5} \)
• B. \( \frac{2 \sqrt{5}}{5} \)
• C. \( \frac{3 \sqrt{5}}{5} \)
• D. \( \frac{4 \sqrt{5}}{5} \)

Answer 1

The Correct Option is C

To find the length of segment \( AC \), we first need to determine the coordinates of the locus point \( C (\alpha, \beta) \). The point \( A \) is given as \( (1, 2) \) and point \( B \) lies on the curve \( x^2 + y^2 = 16 \). The line segment \( AB \) is divided by the point \( P \) in the ratio \( 3:2 \).

The coordinates of point \( B \), given it lies on the circle \( x^2 + y^2 = 16 \), can be parameterized as \( B (\cos \theta, \sin \theta) \) where \( x = 4 \cos \theta \) and \( y = 4 \sin \theta \).

The coordinates of point \( P \) dividing \( AB \) in the ratio \( 3:2 \) is given by section formula:

\[ P \left( \frac{3 \cdot x_2 + 2 \cdot x_1}{3+2}, \frac{3 \cdot y_2 + 2 \cdot y_1}{3+2} \right) \]

Substitute \( A(1, 2) \) and \( B(4 \cos \theta, 4 \sin \theta) \) into the formula:

\[ P \left( \frac{3 \cdot 4 \cos \theta + 2 \cdot 1}{5}, \frac{3 \cdot 4 \sin \theta + 2 \cdot 2}{5} \right) \]

\[ P \left( \frac{12 \cos \theta + 2}{5}, \frac{12 \sin \theta + 4}{5} \right) \]

Since \( C(\alpha, \beta) \) is the center of this locus, the average value of \( \cos \theta \) and \( \sin \theta \) over a complete cycle is 0. Thus, the expected coordinates for the locus center are derived as:

\[ C \left( \frac{2}{5}, \frac{4}{5} \right) \]

Now, find the length \( AC \) using the distance formula:

\[ AC = \sqrt{\left(\frac{2}{5} - 1\right)^2 + \left(\frac{4}{5} - 2\right)^2} \]

\[ AC = \sqrt{\left(\frac{2 - 5}{5}\right)^2 + \left(\frac{4 - 10}{5}\right)^2} \]

\[ AC = \sqrt{\left(\frac{-3}{5}\right)^2 + \left(\frac{-6}{5}\right)^2} \]

\[ AC = \sqrt{\frac{9}{25} + \frac{36}{25}} \]

\[ AC = \sqrt{\frac{45}{25}} \]

\[ AC = \frac{\sqrt{45}}{5} = \frac{3\sqrt{5}}{5} \]

Thus, the length of the line segment \( AC \) is \( \frac{3\sqrt{5}}{5} \).