parabola locus problem If $Q_1$ and $Q_2$ be the angle made by tangents to the axis of $y^2=4x$ from point $P$ and if $Q_1+Q_2=45^{\circ}$ then locus of $P$ is
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Let $P(X,Y)$ be the pole.
The equation of polar or equivalently the chord:
$$Yy-2(x+X)=0 \tag{1}$$
Substitute $x=\dfrac{y^2}{4}$ into $(1)$,
\begin{align}
  y^2-2Yy+2X &= 0 \\
  y_1+y_2 &= 2Y \\
  y_1 y_2 &= 4X \\
  m_1 &= \frac{2}{y_1}  \tag{$2yy'=4$} \\
  m_2 &= \frac{2}{y_2}  \\
  \frac{m_1+m_2}{1-m_1 m_2} &= \frac{2(y_1+y_2)}{y_1 y_2-4} \\
 \tan 45^{\circ} &= \frac{4Y}{4X-4} \\
 Y &= X-1
\end{align}
