show that the common tangent to the parabola `y^(2)=4ax "and" x^(2)=4by` is
`xa^(1//3)+yb^(1//3)+a^(2//3)b^(2//3)=0.`

The equation of the common tangent to y^(2)=4x and 3x^(2) -4y^(2) =12 are

The vertex of the parabola y^(2)+4x-2y+3=0 is

Statement 1: An equation of a common tangent to the parabola y^(2) =16 sqrt3 x and the ellipse 2x^(2) +y^(2) =4 is y= 2x +2sqrt3 . Statement 2: If the line y= mx+ ( 4sqrt3)/( m ) , (m ne 0) is a common tangent to the parabola y^(2)= 16 sqrt3x and the ellipse 2x^(2) +y^(2) =4, then m satisfies m^(4) +2m^(2) =24.

Show that the shortest length of the normal chord to the parabola y^(2) = 4ax is 6sqrt(3a)

Find the equation of the tangent to the parabola x^(2)-4x-8y+12=0" at "(4,(3)/(2))

The equation of the tangent to the cirele x^(2)+y^(2)-2x-4y+3=0" at "(2,3) is

Prove that two parabolas y_(2)=4ax "and" x^(2)=4by intersect (other than the origin ) at an angle of Tan^(-1)[(3a^(1//3)b^(1//3))/(2(a^(2//3)+b^(2//3)))] .

Find the equation of the tangent and normal to the parabola x^(2)-4x-8y+12=0 at (4,(3)/(2))