Suppose a curve whose sub tangent is n times the abscissa of the point of contact and passes through the point (2, 3). Then

Options

A. for n = 1, equation of the curve is `2y = 3x`
B. for n = 1, equation of the curve is `2y^(2) = 9x`
C. for n = 2, equation of the curve is `2y = 3x`
D. for n = 2, equation of the curve is `2y^(2) = 9x`

Answer

A, D

Explanation

If (x, y) is any point on the curve, the sub tangent at (x, y)
`=y(dx)/(dy)`
`therefore" "y(dx)/(dy) = nx" "("given")`
`or" "n(dy)/(y)=(dx)/(x)`
Integrating `n log y = log x + log c`
`or" "log y^(n) = log cx`
`or" "y^(n) = cx" "(i)`
which is the required equation of the family of curves.
Putting `x = 2, y = 3` in (i), we have `3^(n)= 2c or c =(3^(n))/(2)`
Putting this value of c in (i),
`y^(n)=(3^(n))/(2)x`
`or" "2y^(n) = 3^(n) x" "(ii)`
Which is the particular curve passing through the point (2, 3)
Putting n = 1 in (ii), we have 2y = 3x
Which is a straight line
Putting n = 2 in (ii), we have `2y^(2) = 9x`.