Answer to Question #91514 in Analytic Geometry for Ra

Question

Find the equation of the normal to the parabola y²+4x=0 at the point where the line y=x+c touches it.

Answer 1

1. Find inverse functions of the line and the parabola:

$y = x + C ⇒ x = y - C;$

$y^2+4x=0 ⇒x = -\frac{y^2}{4}.$

 

2. The slope of the line with respect to x-axis:

$x = y - C = 1 * y - C;$

$k=1.$

 

3. The tangent to the parabola and the line have the same slope, sо

$x = -\frac{y^2}{4};$

$x'(y) = -\frac{y}{2} = k = 1;$

$y = -2;$

$x = -\frac{y^2}{4} = -\frac{(-2)^2}{4} = -1.$

$C=y-x=-1.$

Thus, the line touches the parabola at the point (-1,-2).

4.The tangent and normal vectors of the line with the slope 'k' are defined by formulas respectively (in reverse order (y,x)):

$T'=(1,k)=(1,1);$

$N'=(-k,1)=(-1,1).$

In general order (x,y) we obtain:

$T=(1,1);$

$N=(1,-1).$

5. Hence, the equation of the normal is

$\frac{x + 1}{1} = \frac{y + 2}{-1};$

$x + 1 = -y - 2;$

$y = -x - 3.$

$Answer: y = -x - 3.$