For the curve `y=4x^3-2x^5,`find all the points at which the tangent passes through the origin.

The tangent to the curve y=xe^(x^2) passing through the point (1,e) also passes through the point

The line 3x+6y=k intersects the curve 2x^2+3y^2=1 at points A and B . The circle on A B as diameter passes through the origin. Then the value of k^2 is__________

The tangent to the curve y=x^2-5x+5. parallel to the line 2y=4x+1, also passes through the point :

Find the point on the curve y=x^3-11 x+5 at which the tangent has the equation y=x-11

Show that the circle x^(2) + y^(2) + 4(x+y) + 4 = 0 touches the coordinates axes. Also find the equation of the circle which passes through the common points of intersection of the above circle and the straight line x+y+2 = 0 and which also passes through the origin.

Find the equation of the normal to the curve x^3+y^3=8x y at the point where it meets the curve y^2=4x other than the origin.

Prove that all points on the curve y^(2)=4a[x+a sin(x/a)] at which the tangent is parallel to the x-axis lie on a parabola.

Find the points on the curve x^(2) + y^(2) – 2x – 3 = 0 at which the tangents are parallel to the x-axis.

Show that all chords of the curve 3x^2-y^2-2x+4y=0, which subtend a right angle at the origin, pass through a fixed point. Find the coordinates of the point.

A line L passing through the point (2, 1) intersects the curve 4x^2+y^2-x+4y-2=0 at the point Aa n dB . If the lines joining the origin and the points A ,B are such that the coordinate axes are the bisectors between them, then find the equation of line Ldot