At any point (x, y) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (– 4, –3). Find the equation of the curve given that it passes through (–2, 1).

_{We know that (x,y) is the point of contact of curve and its tangent.}

slope(m1)for line joining (x,y) and (-4,-3) is

Also we know that slope of tangent of a curve is .

Now, according to the question,

(m2) = 2(m1)

Separating variables,

Integrating both sides,

⇒ log(y + 3) = 2log(x + 4) + log c

Now, this equation passes thorough the point (-2,1).

⇒ 4 = 4c

⇒ c = 1

Substitute the value of c in iii)