Let C be a curve such that the normal at any point P on it meets x-axis and y-axis at A and B respectively. If BP : PA = 1 : 2 (internally) and the curve passes through the point (0, 4), then which of the following alternative(s) is/are correct?

Options

A. The curves passes through `(sqrt(10), -6)`.
B. The equation of tangent at `(4, 4 sqrt(3)) "is 2x" + sqrt(3) y = 20`.
C. The differential equation for the curve is yy' + 2x = 0.
D. The curve represents a hyperbola.

Answer

A, D

Explanation

The equation of normal at `P(x, y) "is"(Y - y)=(-1)/((dy)/(dx))(X-x)`
`therefore" "A(x+y(dy)/(dx), 0) and (0, y+(x)/((dy)/(dx)))`
Now `(1(x+y(dy)/(dx))+2(0))/(1+2)=x rArr x + y (dy)/(dx)=3x`
`rArr" "y(dy)/(dx)=2x`
`rArr" "(y^(2))/(2)=x^(2)+C`
Also (0, 4) satisfy it, so C = 8.
`therefore" "y^(2) = 2x^(2) + 16` (equation of curve)
Which represent a hyperbola

Also `(dy)/(dx)]_(("(4,4 sqrt(3)")))=(2(4))/(4 sqrt(3))=(2)/(sqrt(3))`