If the eccentricity of the curve for which tangent at point `P` intersects the y-axis at `M` such that the point of tangency is equidistant from `M` and the origin is `e ,` then the value of `5e^2` is___

Answer

5

Explanation

`therefore OM=OP`
`y-x(dy)/(dx) = sqrt(x^(2)+y^(2))`
`(dy)/(dx) = y-sqrt(x^(2)+y^(2))/(x)`
`therefore (dy)/(dx) = y/x-sqrt(1+(y/x)^(2))`
Put `y/x=v`or `y=v+x(dv)/(dx)`
`therefore v+x(dv)/(dx) = v-sqrt(1+v^(2))`
`therefore int(dv)/sqrt(1+v^(2))=-int(dx)/x`
`therefore logv+sqrt(1+v^(2))=logc/x`
`therefore y/x+sqrt(1+y^(2)/x^(2))=c/x`
`y+sqrt(x^(2)+y^(2))=c`
Hence, curve is parabola, which has eccentricity 1.