Question

A normal at P (x, y) on a curve meets the x-axis at Q and N is the foot of the ordinate at P. If NQ = $\frac{x(1 + y^{2})}{(1 + x^{2})}$ find the equation of the curve, given that it passes through the point (3, 1) –

• A. 5 (1 + y²) = (1 + x²)
• B. 3 (1 + y²) = (1 + x²)
• C. (1 + y²) = 5 (1 + x²)
• D. None of these

Answer 1

Answer: (A)

5 (1 + y²) = (1 + x²)

Answer 2

In DPNQ,

tan y = $\frac{NQ}{y}$ ; NQ = y

. $\frac{dy}{dx}$

Given that ;

y $\frac{dy}{dx}$ = $\frac{x(1 + y^{2})}{(1 + x^{2})}$

Ž $\frac{ydy}{1 + y^{2}}$ = $\frac{xdy}{1 + x^{2}}$

Integrating $\int_{}^{}\frac{ydy}{1 + y^{2}}$ = $\int_{}^{}\frac{xdy}{1 + x^{2}}$ Ž $\frac{1}{2}$ln (1 + y²) = $\frac{1}{2}$ln

(1 + x²) + c

Ž ln $\sqrt{\frac{1 + y^{2}}{1 + x^{2}}}$ = ln l (where c = ln l)

Ž 1 + y² = l² (1 + x²)

Given that the curve passes through the point (3, 1)

1 + 1 = l² (1 + 9) Ž l² = $\frac{1}{5}$

Required equation is 5 (1 + y²) = (1 + x²)