Question

The shortest length of chord intercepted on a normal to the curve x² -2x-4y+9=0 is

• A. 2√3
• B. 3√2
• C. 6√3
• D. 6√6

Answer 1

Answer: (C)

6√3

Answer 2

The given curve is $x^2 - 2x - 4y + 9 = 0$. Rewrite the equation by completing the square for $x$: $(x^2 - 2x + 1) - 1 - 4y + 9 = 0$ $(x - 1)^2 = 4y - 8$ $(x - 1)^2 = 4(y - 2)$ This is a parabola of the form $X^2 = 4aY$, where $X = x-1$, $Y = y-2$, and $a = 1$.

The shortest length of a normal chord for a parabola $X^2 = 4aY$ is given by the formula $6a\sqrt{3}$. This formula is derived by minimizing the length of the normal chord, which is $L = 4a \frac{(t^2+1)^{3/2}}{t^2}$, where $t$ is the parameter of the point of tangency. The minimum occurs when $t^2 = 2$.

Substituting $a=1$ into the formula for the shortest normal chord length: Shortest length $= 6(1)\sqrt{3} = 6\sqrt{3}$.