The value(s) of a for which two curves `y=ax^(2)+ax+(1)/(24)andx=ay^(2)+ay+(1)/(24)` touch each other is/are

Options

A. `(2)/(3)`
B. `(1)/(3)`
C. `(3)/(2)`
D. `(1)/(2)`

Answer

A, C

Explanation

1,3 The two curves are symmetric about the line y=x.
Hence, they touch each other on y=x.
So, the point of contact is `(alpha,alpha)`.
From any of the equations, we get
`alpha=aalpha^(2)+aalpha+(1)/(24)`
`or24aalpha^(2)+24alpha(a-1)+1=0`
This equation should have Identical roots.
`rArrD=0`
`rArr(24)^(2)(a-1)^(2)-4(24a)=0`
`rArr6a^(2)-13a+6=0`
`rArr(2a-3)(3a-2)=0`
`rArra=3//2,2//3`