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If a curve passes through the point (1, ...

If a curve passes through the point (1, -2) and has slope of the tangent at any point (x,y) on it as `(x^2-2y)/x`, then the curve also passes through the point

A

(a) `(sqrt3,0)`

B

(b) (-1,2)

C

(c) `(-sqrt2,1)`

D

(d) (3,0)

Text Solution

Verified by Experts

The correct Answer is:
(a)
We know that, slope if the tangent at anypoint (x,y) on
the curve is
`dy/dx = (x^(2) -2y)/x` (given)
`rArr dy/dx + 2/xy=x`
shich is a linear differential equation of the form
`dy/dx+P(x)cdot y=Q(x),`
where ` P(x)=2/xand Q(x) = x`
Now, integrating factor
`(IF)=e^(int P(x)dx)=e^(int2/xdx)=e^(2log_(e)x`
`=e^(2log_(e)x^(2) [therefore m log a =log a^(m)]`
`=x^(2) [therefore e^(2log_(e)f(x))=f(x)]`
and the solution of differential Eq.(i) is
`y(IF) = int Q(x)(IF)dx+ CrArry(x^(2))= int xcdotx^(2)dx+C`
`rArr yx^(2) =x^(4)/4+C …(ii)`
`therefore` The curve (ii) passes through the point (1,-2),
therrefore
`-2=1/4+C rArr C= -9/4`
`therefore` Equation of required curve is `4yx^(2)+x^(4)-9.`
Now, checking all the option, we get
only `(sqrt3,0)` satisfy the above equation.
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