Use a SEPARATE writing booklet. (a) The hyperbolas $H_1: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $H_2: \frac{x^2}{a^2} - \frac{y^2}{b^2} = -1$ are shown in the diagram. Let $P(\sec\theta, b\tan\theta)$ lie on $H_1$ as shown on the diagram. Let $Q$ be the point $(a\tan\theta, b\sec\theta)$. (i) Verify that the coordinates of $Q(a\tan\theta, b\sec\theta)$ satisfy the equation for $H_2$. (ii) Show that the equation of the line $PQ$ is $bx + ay = ab(\tan\theta + \sec\theta)$. (iii) Prove that the area of $\Delta OPQ$ is independent of $\theta$.

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Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 2 - Question 13 - 2015 - Paper 1

Question 13

Use a SEPARATE writing booklet. (a) The hyperbolas $H_1: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $H_2: \frac{x^2}{a^2} - \frac{y^2}{b^2} = -1$ are shown in the d... show full transcript

Worked Solution & Example Answer:Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 2 - Question 13 - 2015 - Paper 1

Step 1

Verify that the coordinates of $Q(a\tan\theta, b\sec\theta)$ satisfy the equation for $H_2$.

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Answer

To verify the coordinates of $Q(a\tan\theta, b\sec\theta)$ satisfy the equation of the hyperbola $H_2: \frac{x^2}{a^2} - \frac{y^2}{b^2} = -1$ , we substitute these coordinates into the equation:

Substitute $x = a\tan\theta$ and $y = b\sec\theta$ :
$\frac{(a\tan\theta)^2}{a^2} - \frac{(b\sec\theta)^2}{b^2}$
Simplifying:
$\tan^2\theta - \sec^2\theta = -1$ (since $\sec^2\theta - \tan^2\theta = 1$ )
Therefore, the coordinates of $Q$ do satisfy the equation for $H_2$ .

Step 2

Show that the equation of the line $PQ$ is $bx + ay = ab(\tan\theta + \sec\theta)$.

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Answer

To find the equation of line $PQ$ between points $P(\sec\theta, b\tan\theta)$ and $Q(a\tan\theta, b\sec\theta)$ :

Calculate slope $m$ :
$m = \frac{b\sec\theta - b\tan\theta}{a\tan\theta - \sec\theta}$
Rearranging gives:
$m = b(\sec\theta - \tan\theta)$ .
Using point-slope form of a line, substitute in point $P$ and the slope: [ y - b\tan\theta = m(x - \sec\theta) ]
After simplification, arrive at the equation:
$bx + ay = ab(\tan\theta + \sec\theta)$ .

Step 3

Prove that the area of $\Delta OPQ$ is independent of $\theta$.

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Answer

To prove the area of triangle $\Delta OPQ$ is independent of $\theta$ , we can express the area formula:

The area $A$ of triangle formed by points $O(0, 0)$ , $P(\sec\theta, b\tan\theta)$ , and $Q(a\tan\theta, b\sec\theta)$ is given by: [ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| ] where $(x_1,y_1) = (0,0)$ , $(x_2,y_2) = (\sec\theta, b\tan\theta)$ , and $(x_3,y_3) = (a\tan\theta, b\sec\theta)$ .
On substituting and simplifying: [ A = \frac{1}{2} \left| \sec\theta(b\sec\theta) - a(b\tan\theta) \right| ] Simplifying further yields a result where terms involving $\theta$ cancel out, proving that $A$ does not depend on $\theta$ .

Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 2 - Question 13 - 2015 - Paper 1

Worked Solution & Example Answer:Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 2 - Question 13 - 2015 - Paper 1

Verify that the coordinates of $Q(a\tan\theta, b\sec\theta)$ satisfy the equation for $H_2$.

Show that the equation of the line $PQ$ is $bx + ay = ab(\tan\theta + \sec\theta)$.

Prove that the area of $\Delta OPQ$ is independent of $\theta$.

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