If the line 1 x+m y+n=0 is a normal to the curve x y=c^2 (where c is a constant), then (a) lm<0

step 1 In this question given that this line is a normal to this rectangular hyperbola Then we have to

If the line 1 x+m y+n=0 is a normal to the curve x y=c^2...

Key Concepts

Implicit Differentiation

This technique is used to find the derivative of a function that is given implicitly rather than explicitly. By differentiating both sides of an equation with respect to x (or another variable), one can obtain dy/dx without first solving for y. This is crucial when dealing with curves defined by equations like xy = c², where solving explicitly for one variable in terms of the other might be cumbersome or impossible.

Tangent and Normal Lines

For any differentiable curve, the tangent line at a given point represents the instantaneous direction in which the curve is heading. The normal line, on the other hand, is perpendicular to the tangent line at that point. Understanding the geometric relation between the tangent and the normal lines is essential, because many problems, such as determining if a given line is normal to a curve, rely on this relationship.

Perpendicularity and Slope Relationship

In coordinate geometry, two lines are perpendicular if and only if the product of their slopes is -1, assuming neither line is vertical. This relationship directly connects the slope found via implicit differentiation (which gives the tangent’s slope) with the slope of the normal line, which is the negative reciprocal of the tangent's slope.

Algebraic Conditions for Normality

When determining if a given straight line is normal to a curve, one must equate the slope of the line to the negative reciprocal of the slope of the tangent at the point of contact. This leads to algebraic conditions involving the constants in the line and the curve’s equation. Solving these conditions can often involve substituting the point of contact back into the curve’s equation and verifying consistency.

Properties of Rectangular Hyperbolas

The curve given by xy = c² is a rectangular hyperbola, a type of conic section where the asymptotes are perpendicular. The specific properties of rectangular hyperbolas, such as symmetry and the nature of their tangents and normals, play an important role when solving geometric problems involving normals, as the implicit differentiation and normalization process may yield conditions reflecting these inherent characteristics.

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