Find the value of `n in N`such that the curve `(x/a)^n+(y/b)^n=2`touches the straight line `x/a+y/b=2`at the point `(a , b)dot`A. `(b,a)`B. `(a,b)`C. `(1,1)`D. ` ((1)/(b),(1)/(a)) `

Find the value of `n in N`such that the curve `(x/a)^n+(y/b)^n=2`touch

1.	Find the value of `n in N`such that the curve `(x/a)^n+(y/b)^n=2`touches the straight line `x/a+y/b=2`at the point `(a , b)dot`A. `(b,a)`B. `(a,b)`C. `(1,1)`D. ` ((1)/(b),(1)/(a)) `
Answer» Correct Answer - B We have, `(x^(n))/(a^(n))+(y^(n))/(b^(n)) =2 " "...(i) ` Differentiating with respect to x, we get ` n(x^(n-1))/(a^(n)) +n(y^(n-1))/(b^(n))(dy)/(dx)=0 rArr (dy)/(dx)=-(b^(n))/(a^(n))((x)/(y))^(n-1) ` Suppose the line ` (x)/(a)+(y)/(b)=1 ` touches the curve (i) at `P (alpha ,beta).` The equation of the tangent at `P (alpha, beta )` is ` y- beta =-(b^(n))/(a^(n))((alpha)/(beta))^(n-1) (x-alpha)` ` rArr a^(n) beta^(n-1)y-a^(n)beta^(n)=-b^(n)alpha^(n-1) x+b^(n)alpha^(n)` ` rArr (a^(n)alpha^(n-1))x+(a^(n)beta^(n-1))y=a^(n)beta^(n)+b^(n)alpha^(n) ` ` rArr (a^(n)alpha^(n-1))x+(a^(n)beta^(n-1))y=2a^(n)b^(n) " "[because (alpha,beta) " lies on (i)"] ` Comparing this equation with ` (x)/(a)+(y)/(b)=2 ` , we get ` (ab^(n)alpha^(n-1))/(a^(n)b^(n)) =(ba^(n)beta^(n-1))/(a^(n)b^(n))=(2)/(2) ` ` rArr ((alpha)/(a))^(n-1)=((beta)/(b))^(n-1)=1 ` ` rArr (alpha)/(a)=(beta)/(b)=1 rArr alpha=a,beta =b. `

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