Tangents and it...

For the curve $$x=t^2-1$$, $$y=t^2-t$$, the tangent is perpendicular to $$x$$-axis then

Abscissa of $$p_{1}, p_{2}, p_{3} .... p_{n}$$ are in

If the line $$y=4x-5$$ touches to the curve $${ y }^{ 2 }=a{ x }^{ 3 }+b$$ at the point $$(2,3)$$ then $$7a+2b=$$

If the tangent at $$({x_1},{y_1})$$ to the curve $${x^3} + {y^3} = {a^3}$$ meets the curve again at $$({x_2},{y_2})$$, then

A point on the curve $$y = 2{x^3} + 13{x^2} + 5x + 9$$, the tangent at which passes through the origin is 

The curve that passes through the point $$(2,3)$$ and has the property that the segment of any tangent to it lying between the coordinate axes is bisected by the point of contact, is given by

The tangent to the curve $$x = a(\theta  - \sin \theta );y = a(1 + \cos \theta )$$ at the points $$\theta  = (2k + 1)\pi ,k \in Z$$  are parallel to 

A chord of parabola $${y}^{2}=4ax$$ subtends a right angle at the vertex. The tangents at the extremities of chord intersect on the line:

The slope of the tangent to the curve at a point $$(x,y) $$ on it is proportional to $$(x-2).$$ If the slope of the tangent to the curve at $$(10,-9)$$  on it is $$-3$$. The equation of the curves is .

If $${p_1}$$ and $${p_2}$$ be the lengths of perpendiculars from the origin on the tangent and normal to the curve $${x^\frac{2}{3}} + {y^\frac{2}{3}} = {a^\frac{2}{3}}$$  respectively, then $$4{p_1}^2 + {p_2}^2 = $$