MHTCET 13 Apr 2026 Shift 1 Paper

Concept:For a rectangular hyperbola $xy = -7$, the normal's coefficients $A$ and $B$ have the same sign because the coordinates of any point on the curve have opposite signs.Explanation:The given curve is $xy + 7 = 0$.So, $xy = -7$.Let $\left(x_1, y_1\right)$ be any point on the curve.Then $x_1 y_1 = -7$.Hence $x_1$ and $y_1$ have opposite signs.Differentiate $xy + 7 = 0$ with respect to $x$ using the product rule:$$x \frac{dy}{dx} + y = 0$$$$\frac{dy}{dx} = -\frac{y}{x}$$At $\left(x_1, y_1\right)$, the slope of the tangent is $-\frac{y_1}{x_1}$.So, the slope of the normal is the negative reciprocal:$$\frac{x_1}{y_1}$$The equation of the normal is:$$y - y_1 = \frac{x_1}{y_1}\left(x - x_1\right)$$Multiplying by $y_1$ gives:$$y_1 y - y_1^2 = x_1 x - x_1^2$$Rearranging:$$x_1 x - y_1 y + y_1^2 - x_1^2 = 0$$Comparing with $Ax + By + C = 0$, we get:$$A = x_1, \quad B = -y_1$$Since $x_1$ and $y_1$ have opposite signs, $x_1$ and $-y_1$ have the same sign.Therefore, $A$ and $B$ are either both positive or both negative.Answer:Option A: $A > 0, B > 0$ or $A < 0, B < 0$.