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The graph $y = x^2$ meets the line $y = k$ (where $k > 0$) at points P and Q as shown in the diagram - HSC - SSCE Mathematics Advanced - Question 10 - 2023 - Paper 1

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The graph $y = x^2$ meets the line $y = k$ (where $k > 0$) at points P and Q as shown in the diagram. The length of the interval PQ is L. Let a be a positive number... show full transcript

Worked Solution & Example Answer:The graph $y = x^2$ meets the line $y = k$ (where $k > 0$) at points P and Q as shown in the diagram - HSC - SSCE Mathematics Advanced - Question 10 - 2023 - Paper 1

Step 1

Let a be a positive number.

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Answer

We start with the quadratic equation for the graph of the parabola, which is given as:

y=x2a2y = \frac{x^2}{a^2}

To find the intersections with the line y=ky = k, we set:

k=x2a2k = \frac{x^2}{a^2}

Rearranging this gives:

x2=ka2x=±ka2=±akx^2 = ka^2 \\ x = \pm \sqrt{ka^2} = \pm a\sqrt{k}

The points of intersection are S and T at x=akx = a\sqrt{k} and x=−akx = -a\sqrt{k}.

Step 2

What is the length of ST?

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Answer

The length of ST can be calculated by finding the distance between the points S and T:

ST=(coordinate of S)−(coordinate of T)ST = \text{(coordinate of S)} - \text{(coordinate of T)}
Substituting the values: ST=ak−(−ak)=ak+ak=2akST = a\sqrt{k} - (-a\sqrt{k}) = a\sqrt{k} + a\sqrt{k} = 2a\sqrt{k}

However, since we are asked to express this length in terms of L, we previously established that the length PQ, which corresponds to LL, is:

L=2kL = 2\sqrt{k}

Thus, we can rewrite the ST length in terms of L as follows:

ST=LaST = \frac{L}{a}

Therefore, the correct answer is option C. aLaL.

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