Hyperbola tangent and geometric properties

A question is this type if and only if it involves finding tangents, normals, or geometric properties of a hyperbola parameterized using hyperbolic functions (e.g., x = a cosh t, y = b sinh t).

3 questions · Challenging +1.2

4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials

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Edexcel F3 2014 June Q6

11 marks Challenging +1.2

6. The hyperbola $H$ has equation $$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1$$ The line $l$ is a tangent to $H$ at the point $P ( 4 \cosh \alpha , 2 \sinh \alpha )$, where $\alpha$ is a constant, $\alpha \neq 0$

Using calculus, show that an equation for $l$ is $$2 y \sinh \alpha - x \cosh \alpha + 4 = 0$$ The line $l$ cuts the $y$-axis at the point $A$.
Find the coordinates of $A$ in terms of $\alpha$. The point $B$ has coordinates ( $0,10 \sinh \alpha$ ) and the point $S$ is the focus of $H$ for which $x > 0$
Show that the line segment $A S$ is perpendicular to the line segment $B S$.

Edexcel FP3 2015 June Q6

10 marks Challenging +1.2

The hyperbola $H$ is given by the equation $x ^ { 2 } - y ^ { 2 } = 1$
1. Write down the equations of the two asymptotes of $H$.
2. Show that an equation of the tangent to $H$ at the point $P ( \cosh t , \sinh t )$ is
$$y \sinh t = x \cosh t - 1$$ The tangent at $P$ meets the asymptotes of $H$ at the points $Q$ and $R$.
Show that $P$ is the midpoint of $Q R$.
Show that the area of the triangle $O Q R$, where $O$ is the origin, is independent of $t$.

OCR MEI Further Pure Core 2024 June Q12

12 marks Challenging +1.2

12 The diagram shows the curve with parametric equations $x = 2 \cosh t + \sinh t , y = \cosh t - 2 \sinh t$. \includegraphics[max width=\textwidth, alt={}, center]{83275e7c-7f5a-4f26-b81d-a041e67ac9a2-5_812_808_1283_246}

The curve crosses the positive $x$-axis at A .
1. Determine the value of the parameter $t$ at A , giving your answer in logarithmic form.
2. Find the $x$-coordinate of A , giving your answer correct to $\mathbf { 3 }$ significant figures.
The point B has parameter $t = 0$. Determine the equation of the tangent to the curve at B .