Sol. Suppose the line x cos α + y sin α = p touches the curve x m y n = a m+n at (x 1 , y 1 ). Therefore,

x 1 cos α + y 1 sin α = p ... (i)

and = a m + n ... (ii)

Now,

x m y n = a m+n ⇒ m log x + n log y = (m + n) log a

⇒ + = 0

⇒ = –

⇒ = –

The equation of the tangent to x

m y n = a m+n at (x 1 , y 1 ) is

y – y 1 = – (x –x 1 )

or, (my 1 ) x + (nx 1 ) y = (m + n) x 1 y 1

or, x + y = m + n

If the line x cos α + y sin α = p touches x m y n = a m+n at (x 1 , y 1 ), then this should be same as x cos α + y sin α = p.

∴ = =

⇒ = =

⇒ x 1 = , y 1 =

Substituting the values of m and n in (ii), we get

= a m+n

⇒ m m n n p m+n = (m + n) m+n a m+n (cos α ) m (sin α ) n .

This is the required condition.