Question

A normal is drawn at a point $P(x, y)$ of a curve $y = f(x)$. The normal meets the $X$ axis at $Q$. $l(PQ) = k \cdot$ (k is a constant) Then equation of the curve through $(0, k)$ is

• A. $x^2 + y^2 = k^2$
• B. $(1+k)x^2 + y^2 = k^2$
• C. $x^2 + (1+k^2)y^2 = k^2$
• D. $x^2 + 2y^2 = 2k^2$

Hint:

A curve where the distance from a point to a fixed axis along the normal is constant is a circle.

Answer 1

The Correct Option is A

Step 1: Concept
The length of the normal segment from point $(x, y)$ to the $X$-axis is given by $|y\sqrt{1 + (dy/dx)^2}|$.

Step 2: Meaning
Let the slope of the tangent be $m = dy/dx$. The length $PQ = k$ gives $y^2(1 + m^2) = k^2$.

Step 3: Analysis
Substituting $m = dy/dx$, we get $y^2[1 + (dy/dx)^2] = k^2$. Rearranging, $(dy/dx)^2 = (k^2 - y^2)/y^2$, which leads to the differential equation $y dy = \pm\sqrt{k^2 - y^2} dx$. Integrating both sides results in $x^2 + y^2 = C$.

Step 4: Conclusion
Since the curve passes through $(0, k)$, we find $0^2 + k^2 = C$, so $C = k^2$. The equation is $x^2 + y^2 = k^2$. Final Answer: (A)