(i) Find the locus of a point of the form (e^t+e^-t, e^t-e^-t) (ii) Find the eccentricity of the

step 1 In this problem of conic section, we have to find the locus of a point of the form, say, E to th

(i) Find the locus of a point of the form (e^t+e^-t,...

(i) Find the locus of a point of the form $\left(e^{t}+e^{-t}, e^{t}-e^{-t}\right)$
(ii) Find the eccentricity of the locus
(iii) Find the slope of the tangent drawn to the curve at $\mathrm{t}=1$.

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of an independent parameter. This approach is useful in describing complex curves and in removing the parameter to obtain the locus, which is the relation between the x and y coordinates that defines the curve.

Elimination of Parameters

Elimination of parameters involves removing the independent variable from a set of parametric equations to derive an equation solely in x and y. This process often relies on known identities or algebraic manipulation, enabling one to identify the standard form of the curve, such as a conic section.

Hyperbolic Functions and Their Identities

Hyperbolic functions, such as cosh and sinh, are analogs of trigonometric functions but for hyperbolas. Their fundamental identity, cosh²(t) ? sinh²(t) = 1, is critical for simplifying and recognizing the equations of hyperbolas when the coordinates are expressed in terms of exponential functions.

Eccentricity of a Hyperbola

The eccentricity of a hyperbola is a measure of how much it deviates from being circular. It is defined for a hyperbola in standard form and calculated using the relationship between the transverse and conjugate axes. The eccentricity is always greater than 1 and provides insight into the shape of the hyperbola.

Differentiation of Parametric Equations

Differentiating parametric equations involves computing the derivatives of x and y with respect to the parameter and then forming the ratio dy/dx. This technique is essential for determining the slope of the tangent to the curve at a specific parameter value, thereby providing information about the curve’s local behavior.

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