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Functions, 2ADV F1 2023 HSC 10 MC

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. The graph  \(y=\dfrac{x^2}{a^2}\)  meets the line  \(y=k\)  at points \(S\) and \(T\).

What is the length of \(ST\)?

  1. \(\dfrac{L}{a}\)
  2. \(\dfrac{L}{a^2}\)
  3. \(aL\)
  4. \(a^2L\)
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\(C\)

Show Worked Solution

\(\text{Intersection of}\ \ y=x^2\ \ \text{and}\ \ y=k:\)

\(x^2=k\ \ \Rightarrow\ \ x=\pm \sqrt k\)

\(\therefore L=2\sqrt k\)

\(\text{Intersection of}\ \ y=\dfrac{x^2}{a^2}\ \ \text{and}\ \ y=k:\)

\(\dfrac{x^2}{a^2} \) \(=k\)  
\(x^2\) \(=a^2k\)  
\(x\) \(=\pm a\sqrt k\)  

\(\therefore ST=a \times 2\sqrt k = aL \)

\(\Rightarrow C\)

♦♦♦ Mean mark 24%.

Functions, 2ADV F1 2016 HSC 11e

Find the points of intersection of  `y=-5-4x`  and  `y=3-2x-x^2.`  (3 marks)

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`(4, – 21) and (– 2, 3)`

Show Worked Solution

`y = 3 – 2x – x^2`

`text(Substitute)\ \ y = -5 – 4x\ \ text(into equation)`

`-5 – 4x` `= 3 – 2x – x^2`
`x^2 – 2x – 8` `= 0`
`(x – 4) (x + 2)` `= 0`

  
`:. x = 4 or -2`
 

`text(When)\ \ x = 4,\ \ y = -5 – 4(4) = -21`

`text(When)\ \ x = -2,\ \ y = -5 – 4 (-2) = 3`  
 

`:.\ text(Intersection at)\ \ (4, – 21) and (– 2, 3)`