Application of ...

The slope of the tangent to the curve $$x={t}^{2}+3t-8$$, $$y=2{t}^{2}-2t-5$$ at the point $$(2,-1)$$ is

The line $$y=mx+1$$ is a tangent to the curve $${y}{^2}=4x$$, if the value of $$m$$ is

The abscissa of the points, where the tangent to curve $$y={x}^{3} - 3{x}^{2} - 9x+5$$ is parallel to x-axis, are

The points on the curve $$9{y}^{2}={x}^{3}$$, where the normal to the curve makes equal intercepts with the axes are

Angle between $${ y }^{ 2 }=x$$ and $${ x }^{ 2 }=y$$ at the origin is

If the line $$\alpha\,x+by+c=0$$ is a tangent to the curve $$xy=4$$, then

Let $$y=e^{x^2}$$ and $$y=e^{x^2}\sin\, x$$ be two given curves. Then, angle between the tangents to the curves at any point their intersection is 

The slope at any point of a curve $$y=f\left( x \right) $$ is given by $$\dfrac { dy }{ dx } =3{ x }^{ 2 }$$ and it passes through $$\left( -1,1 \right) $$. The equation of the curve is

Suppose that the equation $$f\left( x \right) ={ x }^{ 2 }+bx+c=0$$ has two distinct real roots $$\alpha $$ and $$\beta $$. The angle between the tangent to the curve $$y=f\left( x \right) $$ at the point $$\left( \dfrac { \alpha +\beta  }{ 2 } ,f\left( \dfrac { \alpha +\beta  }{ 2 }  \right)  \right) $$ and the positive direction of the $$x$$-axis is

The equation of one of the curves whose slope at any point is equal to $$y+2x$$ is