Question

If $P(-5,1)$ is one end of the focal chord $PQ$ of the parabola $x=y^2-8y+2,$ then the slope of the tangent at the other end is

Answer 1

Given parabola is $x=y^2-8y+2$
Tangents at the endpoints of focal chord are perpendicular to each other, so
Slope of tangent at $Q$ is same as slope of normal at $P(-5,1)$
Differentiating the parabola equation,
$1=2y\frac{dy}{dx}-8\frac{dy}{dx}\Rightarrow m=\frac{dy}{dx}=\frac{1}{2(y-4)}$
Slope of normal at $P$ is
$m_n=-\frac{1}{m}=-2(y-4)\Rightarrow m_n=-2(1-4)=6$