Tangents and it...

If the circle $$x^2+y^2+2gx+2fy+c=0$$ is touched by $$y=x$$ at P such that OP = $$6\sqrt{2}$$
then the value of c is

The angle made by the tangent of the curve $$\displaystyle x = a(t + \sin t \cos t); y = a (1 + \sin t)^2$$ with the x-axis at any point on it is

Equation of the line through the point $$\left(\dfrac{1}{2}, 2 \right)$$ and tangent to the parabola $$\displaystyle y = \frac {-x^2}{2}+2$$ and secant to the curve $$\displaystyle y = \sqrt {4 - x^2}$$ is :

What is the minimum intercept made by the axes on the tangent to the ellipse $$ \displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2}=1$$ ?

Let $$C$$ be the curve $$\displaystyle y = x^3$$ (where $$x$$ takes all real values.) The tangent at $$A$$ meets the curve again at $$B$$. If the gradient of the curve at $$B$$ is $$K$$ times the gradient at $$A$$ then $$K$$ is equal to

The graphs $$y=2x^3-4x+2$$ and $$y=x^3+2x-1$$ intersect at exactly 3 distinct points. The slope of the line passing through two of these points

Consider the curve represented parametrically by the equation
$$\displaystyle x = t^3 - 4t^2 - 3t$$ and $$\displaystyle y = 2t^2 + 3t - 5$$ where $$\displaystyle t \: \epsilon \: R$$.
If $$H$$ denotes the number of point on the curve where the tangent is horizontal and $$V$$ the number of point where the tangent is vertical then

The number of points on the curve $$x^{3/2}+y^{3/2}=a^{3/2}$$, where the tangents are equally inclined to the axes, is

Let $$\displaystyle f(x) = ln \: mx (m > 0)$$ and $$g(x) = px$$. Then the equation $$\displaystyle |f(x)| = g(x)$$ has only one solution for

A point $$\displaystyle P(a, a^n)$$ on the graph of $$\displaystyle y = x^n (n \: \epsilon \: N)$$ in the first quadrant a normal is drawn. The normal intersects the y-axis at the point $$(0, b)$$. If $$\displaystyle _{a \rightarrow 0}^{Lim \: b} \textrm{=} \frac {1}{2}$$, then n equals