Question

The line L given by $straight x over 5 space plus straight y over straight b space equals space 1$ passes through the point (13, 32). The line K is parallel to L and has the equation $straight x over straight c space plus straight y over 3 space equals space 1$ . Then the distance
between L and K is

$fraction numerator 23 over denominator square root of 15 end fraction$
$square root of 17$
$fraction numerator 17 over denominator square root of 15 end fraction$
$fraction numerator 23 over denominator square root of 17 end fraction$

Solution

fraction numerator 23 over denominator square root of 17 end fraction

Since, the line L is passing through the point (13,32)
Therefore,
$13 over 5 space plus 32 over straight b space equals space 1 rightwards double arrow space 32 over straight b space equals space minus 8 over 5 rightwards double arrow space straight b space equals space minus 20$
The line K is parallel to the line L then its equation must be
$straight x over 5 minus straight y over 20 space equals straight a space Or fraction numerator straight x over denominator 5 straight a end fraction minus fraction numerator straight y over denominator 20 straight a end fraction space equals space 1 On space comparing space with space straight x over straight c space plus straight y over 3 space equals space 1 comma space we space straight g space et 20 straight a space equals space minus space 3 comma space straight c space equals 5 straight a space equals space minus 3 over 4 Hence comma space the space distance space between space lines fraction numerator vertical line straight a minus 1 vertical line over denominator square root of begin display style 1 over 25 plus 1 over 400 end style end root end fraction space equals space fraction numerator open vertical bar begin display style fraction numerator negative 3 over denominator 20 end fraction minus 1 end style close vertical bar over denominator square root of begin display style 17 over 400 end style end root end fraction space equals space fraction numerator 23 over denominator square root of 17 end fraction$

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