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A family of lines through a fixed point

A family of lines through a fixed point

IP 1
7 marks

For each value of the constant \(k\), the line \(L_k\) has equation \(y=k(x-3)+5\).

  1. Show that every line \(L_k\) passes through the point \((3,5)\), and state the gradient of \(L_k\).[2]
  2. Find the value of \(k\) for which \(L_k\) passes through the origin.[2]
  3. For \(k\ne0\), find the \(x\)-intercept of \(L_k\) in terms of \(k\). Explain why \(L_0\) has no \(x\)-intercept.[3]

Solution:

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Answer:(a) at \(x=3\), \(y=5\) for every \(k\); gradient \(k\) (b) \(k=\dfrac53\) (c) \(x=3-\dfrac5k\); \(L_0\) is \(y=5\), parallel to the \(x\)-axis

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