Let u and v be two parallel lines passing through the points A=(5,0) and

B=(-5,0) respectively.Let the line 4x+3y=25 meet u at P and v at Q.

a)If he length of PQ is 5 units, show that here are two possibilities for the pair of parallel lines u and v.

b)Write down the equations of all four lines determined above.

c)Find the equations of the diagonals of the parallelogram formed by these four lines and find the area of the above parallelogram.

Answer only part c.

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The equations of two pairs of parallel lines, as obtained from part b) of the question, are:

`y=4/7(x-5)` --- (i)

`y=-4/13(x-5)` --- (ii)

`y=4/7(x+5)` --- (iii)

`y=-4/13(x+5)` --- (iv)

Upon solving, the points of their interceptions are:

(1.5, -2), (5, 0), (-1.5, 2) and (5, 0) respectively.

These are the four corners of the parallelogram formed by the four lines.

The diagonals can be obtained by connecting alternate points. Applying two point form of the equation of a line, the equations of the diagonals are:

`(y+2)=(2+2)/(-1.5-1.5) (x-1.5)`

`rArr y+2=-2/3(2x-3)`

`rArr 4x+3y=0` --- (v)

And,

`(y-0)=(0-0)/(-5-5) (x-5)`

`rArr y=0` --- (vi)

Length of diagonal 1 = `sqrt((-2-2)^2+(-1.5-1.5)^2)=sqrt(4^2+3^2)=5` units

Length of diagonal 2 = `sqrt((0-0)^2+(-5-5)^2)=sqrt(10^2)=10` units

Area of the parallelogram `=1/2 *d_1*d_2=1/2 *5*10=25` sq. units.

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