Amit M Agarwal Solutions for Chapter: dy/dx as a Rate Measurer & Tangents, Normals, Exercise 1: Target Exercise 7.1

The equation of the tangents to the curve $(1+x^2)y=1$ at the points of its intersection with the curve $(1+x)y=1$, is given by

The tangent lines for the curve $y={\int }_0^{x}2\left|t\right|dt$ which are parallel to the bisector of the first quadrant angle, is given by

 The equation of normal to $x+y=x^y$, where it intersects $x$-axis, is given  by:

The equation of normal at any point $\theta$ to the curve $x=a\cos \theta +a\theta \sin \theta , y=a\sin \theta -a\theta \cos \theta$, $a>0$ is always at a distance of

If the tangent 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 

The area bounded by the axes of reference and the normal to $y={\log }_{e}x$ at $(1, 0)$, is 

If $\frac{x}{a}+\frac{y}{b}=2$ touches the curve $\frac{x^n}{a^n}+\frac{y^n}{b^n}=2$ at the point $(\alpha , \beta )$ and $n, a, b\ne 0$, then

The equation of tangents to the curve $y=\cos \left(x+y\right), -2\pi \le \mathrm{x}\le 2\pi$ that are parallel to the line $x+2y=0$, is: