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)?
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.