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

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: \[ (x^2 + y^2 -1)(x^2 + y^2 - 2x - 2y + 1) = 0, x \neq 0,1. \]