Amit M Agarwal Solutions for Chapter: dy/dx as a Rate Measurer & Tangents, Normals, Exercise 4: Proficiency in 'dy/dx as a Rate Measurer & Tangents, Normals' Exercise 2

Show that the angle between the tangents at any point $P$ and the line joining $P$ to the origin '$O$' is the same at all points of the curve $\mathrm{l}\mathrm{o}\mathrm{g}\left(x^2+y^2\right)=c\mathrm{t}\mathrm{a}{\mathrm{n}}^{-1}\left(y/x\right)$, where $c$ is constant.

If the equation of two curves is $y^2=4ax$ and $x^2=4ay$. Find the angle of intersection of two curves.

If the equation of two curves is $y^2=4ax$ and $x^2=4ay$, find the equation of common tangents to these curves.

A straight line intersects the three concentric circles at $A,B,C$. If the distance of the line from the centre of the circles is $P$, prove that the area of the triangle formed by the tangents to the circle at $A,B,C$ is $\left(\frac{1}{2P}·AB·BC·CA\right)$.

Find the equation of all possible curves such that length of intercept made by any tangent on $x-$axis is equal to the square of $x-$coordinate of the point of tangency. Given that the curve passes through $\left(\mathrm{2,} 1\right)$.

The tangent to the curve $y=x-x^3$ at a point $P$ meets the curve again at $Q$. Prove that one point of trisection of $PQ$ lies on the $y-$axis. Also find the locus of the other points of trisection.

Determine all the curves for which the ratio of the length of the segment intercepted by any tangent on the $y$-axis to the length of the radius vector is constant.

If $t$ is a real number satisfying $2t^3-9t^2+30-a=0$, then the values of the parameter $a$ for which the equation $x+\frac{1}{x}=t$ gives six real and distinct values of $x$.