consider a family of curves,where the ordinates is proportional to the cube of the abscissa and let A be a fixed point in a plane,which has cordinates (a,b).ith the reference to the rectangular cordinate  axes.

 

(Q) if tangents be drawn through A to the members of the familiy of the curves then the locus of the points of contact is

 

 

(a)xy+bx-3ay=0                                       (b)xy-4bx+3ay=0

 

(c)2xy+bx-3ay=0                                      (d)2xy-4bx+3ay=0

 

 

(Q)if normals be drawn through A to the members of the family of the curves then the feet of these normals to the curves also lie on the curve

 

(a)xy+bx-3ay                                             (b)xy-4bx+3ay=0

 

 

(c)x^2-3y^2=ax-3by                                    (d)x^2+3y^2=ax+3by

 

 

 

(Q)if the tangent through A to a curve cuts the curve again at a point B then the locus of the point B is

 

 

(a)xy-4bx+3ay=0                                         (b)2xy+bx-3ay=0

 

(c)x^2-3y^2=ax-3by                                      (d)a^2x^2+b^2y^2=1

 

 

where x^2 has meaning square

q1) Let the pt of contact be (h,k), we have to find the locus of THIS point.

 

The curve is of the form y = kx³

 

.: slope of tangent at (h,k) is 3Kh²

en of tangent = 3kh² (x-h) = y-k

 

K = k/h³

 

.: en of tangent is  3 (k/h³)h² (x-h) = y-k

 

2kh+bk=3ak

 

Hence locus of h,k is 2xy+bx-3ay=0