Question 3 (15 marks) Use a SEPARATE writing booklet.. (a) The diagram shows the graph of $y = f(x)$. Draw separate one-third page sketches of the graphs of the following: (i) $y = -\frac{1}{\sqrt{x}}$ (ii) $y^2 = f(x)$ (iii) $y = |f(x)|$ (iv) $y = \ln(f(x))$. Question 3 (continued) (b) The distinct points $P\left(c_{p}, \frac{c}{p}\right)$ and $Q\left(c_{q}, \frac{c}{q}\right)$ are on the same branch of the hyperbola $\mathcal{H}$ with equation $xy = c^2$. The tangents to $\mathcal{H}$ at $P$ and $Q$ meet at the point $T$. (i) Show that the equation of the tangent at $P$ is $x + p y = 2c p$. (ii) Show that $T$ is the point $\left( \frac{2c p q}{p + q}, \frac{2c}{p + q} \right)$. (iii) Suppose $P$ and $Q$ move so that the tangent at $P$ intersects the x-axis at $(c_{q}, 0)$. Show that the locus of $T$ is a hyperbola, and state its eccentricity.

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Question 3 (15 marks) Use a SEPARATE writing booklet. - HSC - SSCE Mathematics Extension 2 - Question 3 - 2002 - Paper 1

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Question 3 (15 marks) Use a SEPARATE writing booklet.. (a) The diagram shows the graph of $y = f(x)$. Draw separate one-third page sketches of the graphs of the f... show full transcript

Worked Solution & Example Answer:Question 3 (15 marks) Use a SEPARATE writing booklet. - HSC - SSCE Mathematics Extension 2 - Question 3 - 2002 - Paper 1

Step 1

Draw separate one-third page sketches of the graphs of the following: (i) $y = -\frac{1}{\sqrt{x}}$

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Answer

To sketch the graph of $y = -\frac{1}{\sqrt{x}}$ , note that:

The domain of the function is $x > 0$ , as square roots are defined only for non-negative values.
As $x$ approaches $0$ from the right, $y$ approaches $- rac{1}{\sqrt{0}}$ which tends to $-\\infty$ .
As $x$ increases, $y$ approaches $0$ (but remains negative).

The graph will lie entirely below the x-axis, approaching the x-axis asymptotically as $x$ increases.

Step 2

Draw separate one-third page sketches of the graphs of the following: (ii) $y^2 = f(x)$

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Answer

The graph of $y^2 = f(x)$ utilizes the original graph of $y = f(x)$ , but reflects it about the x-axis. It is essential to:

Identify the range of $f(x)$ which may be negative.
Draw the positive values of $y$ based on $f(x)$ and reflect any negative values to the negative $y$ -axis, resulting in two branches.

Step 3

Draw separate one-third page sketches of the graphs of the following: (iii) $y = |f(x)|$

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Answer

To sketch $y = |f(x)|$ , perform the following steps:

Plot the graph of $f(x)$ as given.
Identify portions of the graph that are below the x-axis. These segments should be reflected above the x-axis.
Retain the segments above the x-axis as they are. The resulting graph will be entirely non-negative.

Step 4

Draw separate one-third page sketches of the graphs of the following: (iv) $y = \ln(f(x))$

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Answer

For the graph of $y = \ln(f(x))$ , consider these important points:

The function is defined only where $f(x) > 0$ .
Identify the x-values for which $f(x) = 1$ ; these will correspond to $y = 0$ .
As $f(x)$ increases, $\\ln(f(x))$ will also increase.
As $f(x)$ approaches zero, the logarithmic function will tend to $-\infty$ . Ensure to mark these critical points clearly on the graph.

Step 5

Show that the equation of the tangent at P is $x + p y = 2c p$. (i)

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Answer

To find the equation of the tangent line at point $P\left(c_{p}, \frac{c}{p}\right)$ on the hyperbola $xy = c^2$ :

The derivative of $y = \frac{c^2}{x}$ gives the slope, $m = -\frac{c^2}{x^2}$ .
Evaluate the slope at $P$ : $m_P = -\frac{c^2}{(c_p)^2}$ .
Using point-slope form, with point $P$ and slope $m_P$ , formulate the equation: $y - \frac{c}{p} = -\frac{c^2}{(c_p)^2}(x - c_p)$
Rearranging yields the equation of the tangent: $x + p y = 2c p$ .

Step 6

Show that T is the point \left( \frac{2c p q}{p + q}, \frac{2c}{p + q} \right). (ii)

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Answer

To find the coordinates of point $T$ , the intersection of the tangents at points $P$ and $Q$ :

Write the equations of the tangents at points $P$ and $Q$ using the forms derived previously.
Set these two equations equal to solve for $x$ and $y$ .
After substituting the necessary expressions of $p$ and $q$ , determine the coordinates at $T$ as $\left( \frac{2c p q}{p + q}, \frac{2c}{p + q} \right)$ .

Step 7

Show that the locus of T is a hyperbola, and state its eccentricity. (iii)

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Answer

To prove that the locus of $T$ is a hyperbola:

Substitute the coordinates found for point $T$ into the relationship derived from the hyperbola's equation.
Manipulate the terms to bring it into the standard hyperbola form: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ where $a$ and $b$ are expressions derived from your coordinates.
The eccentricity $e$ of the hyperbola can be derived from the relationship $e = \sqrt{1 + \frac{b^2}{a^2}}$ , where terms for $a$ and $b$ are referenced from your manipulations.

Question 3 (15 marks) Use a SEPARATE writing booklet. - HSC - SSCE Mathematics Extension 2 - Question 3 - 2002 - Paper 1

Worked Solution & Example Answer:Question 3 (15 marks) Use a SEPARATE writing booklet. - HSC - SSCE Mathematics Extension 2 - Question 3 - 2002 - Paper 1

Draw separate one-third page sketches of the graphs of the following: (i) $y = -\frac{1}{\sqrt{x}}$

Draw separate one-third page sketches of the graphs of the following: (ii) $y^2 = f(x)$

Draw separate one-third page sketches of the graphs of the following: (iii) $y = |f(x)|$

Draw separate one-third page sketches of the graphs of the following: (iv) $y = \ln(f(x))$

Show that the equation of the tangent at P is $x + p y = 2c p$. (i)

Show that T is the point \left( \frac{2c p q}{p + q}, \frac{2c}{p + q} \right). (ii)

Show that the locus of T is a hyperbola, and state its eccentricity. (iii)

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