If, A normal is drawn at a point P(x, y) of a curve. It meets the x-axis at Q. If PQ is of constant length k. What kind of curve is passing through (0, k)?

(a) Parabola

(b) Hyperbola

(c) Ellipse

(d) Circle

I had been asked this question in my homework.

This intriguing question comes from Linear First Order Differential Equations topic in portion Differential Equations of Mathematics – Class 12

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The correct option is (d) Circle

The explanation is: Equation of the normal at a point P(x, y) is given by

Y – y = -1/(dy/dx)(X – x) ….(1)

Let the point Q at the x-axis be (x1 , 0).

From (1), we get

y(dy/dx) = x1 – x ….(2)

Now, giving that PQ^2 = k^2

Or, x1 – x + y^2 = k^2

=>y(dy/dx) = ± √(k^2 – y^2) ….(3)

(3) is the required differential equation for such curves,

Now solving (3) we get,

∫-dy/√(k^2 – y^2) = ∫-dx

Or, x^2 + y^2 = k^2 passes through (0, k)

Thus, it is a circle.