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202212141214 Solution to 1999-CE-AMATH-II-5

A family of straight lines is given by the equation

y-3+k(x-y+1)=0,

where k is real.

(a) Find the equation of a line L_1 in the family whose x-intercept is 5.
(b) Find the equation of a line L_2 in the family which is parallel to the x-axis.
(c) Find the acute angle between L_1 and L_2.

Roughwork.

(a)

Substituting (5,0) for the x-intercept:

\begin{aligned} (0)-3+k((5)-(0)+1) & = 0 \\ k & = \frac{1}{2} \\ \Longrightarrow \enspace L_1: \big\{ x+y-2 &=0 \big\} \\ \end{aligned}

(b)

Differentiating on both hand sides,

\begin{aligned} \mathrm{d}y+k(\mathrm{d}x-\mathrm{d}y) & =0 \\ \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{k}{k-1} & = 0 \\ k & = 0 \\ \Longrightarrow \enspace L_2: \big\{ y & = 3 \big\} \\ \end{aligned}

(c) 45^\circ.

This problem is not to be attempted.