Question

If the line segment joining the points \( (1,0) \) and \( (0,1) \) subtends an angle of \( 45^\circ \) at a variable point \( P \), then the equation of the locus of \( P \) is: 

• A. \( (x^2 + y^2 -1)(x^2 + y^2 - 2x - 2y + 1) = 0, x \neq 0,1 \)
• B. \( (x^2 + y^2 -1)(x^2 + y^2 + 2x + 2y + 1) = 0, x \neq 0,1 \)
• C. \( x^2 + y^2 + 2x + 2y + 1 = 0 \)
• D. \( x^2 + y^2 = 4 \)

Hint:

To determine the locus of a point subtending a fixed angle at a given segment, use the standard angle subtended formula for chords and simplify accordingly.

Answer 1

The Correct Option is A

To find the equation of the locus of \( P \) such that the line segment joining points \( (1,0) \) and \( (0,1) \) subtends an angle of \( 45^\circ \), we can use the concept of circle power and angle subtension.

The general concept is that if a line segment \( AB \) subtends angle \(\theta\) at point \( P \), then the point \( P \) lies on a circle known as the circle of Apollonius with \( AB \) as a chord for given angle \(\theta\).

Given points are \( A(1,0) \) and \( B(0,1) \). Using the formula for angle subtended by a chord at any point, we know that if \( APB = 45^\circ\), then point \( P(x, y) \) lies on a combination of two lines or a circle through a specific configuration.

The condition for angle \(\theta\) at \( P \) is given by the equation:

\( \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1m_2} \right| \)

Here \(\theta = 45^\circ\), hence: \( m_1 \cdot m_2 = -1 \) (the product of slopes must be -1 for perpendicularity, which is directly related to \( 45^\circ\)).

The locus of point \( P \) is given by the equation of the radical axis which is derived from two circles:

- The first circle is centered at \( A(1,0) \) with arbitrary radius: \( (x-1)^2 + y^2 = r^2 \).

- The second circle is centered at \( B(0,1) \) with same radius: \( x^2 + (y-1)^2 = r^2 \).

The difference between these two equations gives us the radical axis:

\(2x - 2y = 1\)

Considering the path of the point \( P \) on the plane making \( 45^\circ \):

The circle centers, which are points of equal scaled distance from \( A \) and \( B \), give:
\( (x^2 + y^2 - 1) \) and \( (x^2 + y^2 - 2x - 2y + 1) = 0 \)

This results in two possibilities where either circle condition holds:

\( (x^2 + y^2 -1)(x^2 + y^2 - 2x - 2y + 1) = 0 \)

Thus the correct locus equation is:

\((x^2 + y^2 -1)(x^2 + y^2 - 2x - 2y + 1) = 0\)

Additionally, \( x \neq 0, 1 \) as they would invalidate the range of possible lines considered.

Answer 2

We need to determine the equation of the locus of the point \( P(x,y) \) such that the line segment joining the points \( A(1,0) \) and \( B(0,1) \) subtends an angle of \( 45^\circ \) at \( P \). 
Step 1: Use the Angle Subtended Formula 
The general condition for a chord subtending a given angle at a point is given by: \[ \tan^2 \theta = \frac{4(ab - h^2 - k^2)}{(a^2 + b^2 + 2h a + 2kb)} \] where \( (h,k) \) is the point \( P(x,y) \) and the given chord endpoints are \( A(1,0) \) and \( B(0,1) \). 
Step 2: Substitute Given Values 
For \( 45^\circ \), we know \( \tan^2 45^\circ = 1 \). Using this equation, we simplify and derive the locus equation.
Step 3: Obtain the Required Equation 
After simplifying, we obtain: \[ (x^2 + y^2 -1)(x^2 + y^2 - 2x - 2y + 1) = 0, x \neq 0,1. \]