Let $(x_0,y_0)$, where $y_o \neq 0$, be a point on the parabola $y^2=2px$. Find an equation of the tangent line to the parabola passing through $(x_0,y_0)$.
I did my work for this, but I cannot get the answer, which my tutor swears on his job (Joking, but I'm pretty sure its correct), that is correct.
Given Answer: $y_0y=p(x+x_0)$
Here's my workings.
Differentiate with respect to $x$, we have $2y\frac{\mathrm{dy} }{\mathrm{d} x}=2p$ Therefore, $\frac{\mathrm{dy} }{\mathrm{d} x}=\frac {p} {y}$
At Point $(x_0,y_0), \frac{\mathrm{dy} }{\mathrm{d} x}=\frac {p} {y_0}$
Using the equation for straight lines, we have $(y-y_0)=\frac{p}{y_0}(x-x_0)$ $yy_0-y_0^2=px-px_0$
By now, it should be clear I will not get the answer. Did I make any mistake?
I reverse-engineered my tutor's answer and I realised he used $(y-y_0)=\frac{p}{y}(x-x_0)$ instead. $y^2-yy_0=px-px_0$
$yy_0=px_0-px+y^2$
$yy_0=px_0-px+2px$
$yy_0=px_0+px=p(x+x_0)$
Why is it that when I use $\frac {p}{y}$ versus $\frac {p}{y_0}$, both gives me the same answer? Which should I be using?