Question

The normal lines to a given curve at each point (x, y) on the curve pass through (2, 0). The curve passes through (2, 3). Find the equation of the curve.

Answer 1

Let P(x, y) be a point on the curve y = f(x). Then slope of the normal to the curve is \(-\frac{1}{(\frac{dy}{dx})}\)

∴ equation of the normal is

Since, this normal passes through (2,0), we get

Integrating both sides, we get

This is the general equation of the curve. Since, the required curve passed through the point (2, 3), we get

2² + 3² = 4(2) + c

∴ c = 5

∴ equation of the required curve is x² + y² 

= 4x + 5.