NIMCET 2024 — Mathematics PYQ

To find the condition for the real number $a$, we must find the slope of the normal to the given curve and relate it to the slope of the given line.

Step 1: Find the slope of the given line

The equation of the given line is:

$$a^{2}x+ay+1=0$$

Rewriting it in the slope-intercept form ($y=mx+c$):

$$ay=-a^{2}x-1$$

$$y=-\left(\frac{a^2}{a}\right)x-\frac{1}{a}$$

$$y=-ax-\frac{1}{a}(\text{assuming }a\ne 0)$$

Thus, the slope of the given line ($m_L$) is:

$$m_L=-a$$

Step 2: Find the slope of the normal to the curve

The equation of the curve is a rectangular hyperbola:

$$xy=1⟹y=\frac{1}{x}$$

Differentiating both sides with respect to $x$ to find the slope of the tangent ($m_T$):

$$\frac{dy}{dx}=-\frac{1}{x^2}$$

The slope of the normal ($m_N$) at any point $(x,y)$ on the curve is perpendicular to the tangent, so $m_N=-\frac{1}{m_T}$:

$$m_N=-\frac{1}{-\frac{1}{x^2}}=x^2$$

Step 3: Equate the slopes

Since the given line is normal to the curve, its slope must equal the slope of the normal at some point on the curve:

$$m_L=m_N$$

$$-a=x^2$$

Step 4: Analyze the condition for real number $a$

Since $x$ is a coordinate of a point on the curve, $x$ must be a real number. For any real number $x$, the square of the number is always positive (x^2 > 0, since $x\ne 0$ for the curve $xy=1$).

Therefore:

x^2 > 0

-a > 0

Multiplying both sides by $-1$ flips the inequality sign:

a < 0

Final Answer

The correct option is A (a < 0).