Mathematics · Ch 5 — Two Dimensional Analytical Geometry-II

Equation of Tangent and Normal to the Parabola y² = 4ax

5.6.1

Tangent, cartesian form. For two points $P(x_1,y_1),Q(x_2,y_2)$ on $y^2=4ax$ : $y_1^2=4ax_1$ and $y_2^2=4ax_2$ , so $y_1^2-y_2^2=4a(x_1-x_2)$ , giving the chord's slope $\dfrac{y_1-y_2}{x_1-x_2}=\dfrac{4a}{y_1+y_2}$ . Letting $Q\to P$ (so $y_2\to y_1$ ) turns this chord into the tangent at $P$ , with slope $\dfrac{2a}{y_1}$ : $y-y_1=\frac{2a}{y_1}(x-x_1) \implies yy_1=2a(x+x_1).$
Tangent, parametric form. At the point $(at^2,2at)$ , substituting $x_1=at^2,y_1=2at$ into $yy_1=2a(x+x_1)$ and simplifying gives $\boxed{yt=x+at^2}$ .
Normal, cartesian form. The tangent's slope is $2a/y_1$ , so the normal's slope is $-y_1/2a$ : $y-y_1=-\frac{y_1}{2a}(x-x_1) \implies xy_1+ay_1=x_1y_1+ay_1^{3}/y_1 \implies xy_1+2ay=x_1y_1+2ay_1.$
Normal, parametric form. At $(at^2,2at)$ , the cartesian normal simplifies (using $y_1=2at,x_1=at^2$ ) to $\boxed{y+xt=2at+at^3}$ . …