Tangents and it...

The normal to the curve, $${x}^{2}+2xy-{3y}^{2}=0,\ at\left (1,1\right)$$:

If the tangent at P of the curve $$y^2=x^3$$ intersects the curve again at Q and the straight lines OP, OQ ma angles $$\alpha, \beta$$ with the x-axis where 'O' is the origin then $$\tan\alpha/\tan\beta$$ has the value equal to?

If $$x={t}^{2}$$ and $$y=2t$$, then equation of the normal at $$t=1$$ is

The area of the triangle formed by the coordinate axes and a tangent to the curve $$xy={a}^{2}$$ at the point $$({x}_{1},{y}_{1})$$ is

The equation of the tangent to curve $$\sqrt{\frac{x}{a}}+\sqrt{\frac{y}{b}}=2$$ at the point (a, b) is 

The equation of tangent to the ellipse $$4{x^2} + 9{y^2} = 36$$ at $$(3, - 2)\,is\,$$

The equation of tangents to the ellipse $${x^2} + 4{y^2} = 25$$ at the point whose ordinate is 2, is 

The equation of the normal to the curve $${x}^{4}=4y$$ through the point $$(2,4)$$ is

Number of tangents drawn from the point $$\left (-1/2,0\right)$$ to the curve $$y={e}^{x}$$. (Here { } denotes fractional part function ). 

The equation of tangent to the curve $$y=x^{2}+4x+1$$ at $$\left(-1,-2\right)$$ is