Problem related to tangents and normals of a curve.

Question

I 've been trying the both sums, while first one I've no clue how to start about in the second one I a getting stuck.

[1] The equation of the normal at any point $\theta$ on the curve $x=a \cos\theta+ a\theta \sin\theta$, $y =a\sin \theta - a\theta \cos\theta$ is always at a distance

(a) 2a unit from origin
(b) a unit from origin (c) $\frac{1}{2}$ unit from origin (d) None of these . And (b) is the correct answer given. It is a MCQ.

[2] If the tangent line at $(x_0,y_0)$ to the curve $x^3+y^3=a^3$ meets the curve again at $(x_1,y_1)$ , then $\frac {x_1}x_0+\frac{y_1}y_0$ is equal to what?

Answer 1

for the first one we need to find $dy/d\theta $ and then $dx/d\theta $. then we can exploit the fact that

$dy/dx$ = ($dy/d\theta $)/$(dx/d\theta $).

that gives the slope of the tangent ,

Now we know that the normal slope = $(-1)/(tangent slope) $.

now the pt $x=a \cos\theta+ a\theta \sin\theta$, $y =a\sin \theta - a\theta \cos\theta$ ,

We can use point slope formula to get the normal equation. . then we use some coordinate geometry basics that state that

perpendicular distance of $ax+by+c=0 $ from any point (x1, y1) =

          [mod{(a(x1)+b(y1)+c}]/ $\sqrt\{a^2 + b^2}$

given here that $(x1, y1) $ = (0,0) and the perpendicular distance is c , just substitute and you are done . (done assuming the fact that you know the basics of differentiation and coordinate geometry)