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

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Aicharya · Accepted Answer

Slope of $\mathrm{AP}=\frac{k}{h-1}$ Slope of $\mathrm{BP}=\frac{k-1}{h} \therefore \tan \theta=\left\lvert\, \frac{m_1-m_2}{1+m_1 m_2}\right\lvert$ $\Rightarrow \tan 45^{\circ}=\left|\frac{\frac{k}{h-1}-\frac{k-1}{h}}{1+\frac{k(k-1)}{h(h-1)}}\right| \Rightarrow 1=\left|\frac{k h-(h-1)(k-1)}{h(h-1)+k(k-1)}\right|$ $\therefore h^2-h+k^2-k=k h-k h+h+k-1$ $\Rightarrow h^2+k^2-2 h-2 k+1=0$ $\therefore$ Locus is $x^2+y^2-2 x-2 y+1=0$ $\because(1,0)$ and $(0,1)$ satisfy $x^2+y^2-1=0$ $\therefore$ Locus is $\left(x^2+y^2-1\right)\left(x^2+y^2-2 x-2 y+1\right)=0$

If the line segment joining the points $(1,0)$ and $(0,1)$ subtends an angle of $45^{\circ}$ at a variable…

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