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The line L given by $\dpi{100} \frac{x}{5}+\frac{y}{b}=1$  passes through the point (13, 32). The line K is parallel to L and has the equation $\dpi{100} \frac{x}{c}+\frac{y}{3}=1.$ Then the distance between L and K is

• Option 1)

$\frac{23}{\sqrt{15}}\;$

• Option 2)

$\; \; \sqrt{17}\;$

• Option 3)

$\; \frac{17}{\sqrt{15}}\;$

• Option 4)

$\; \frac{23}{\sqrt{17}}$

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As we learnt in

Condition for parallel lines -

$m_{1}= m_{2}$

- wherein

Here $m_{1},m_{2}$ are the slope of two lines

and

Distance between two parallel lines -

$\rho =\frac{\left | c_{2}-c_{1} \right |}{\sqrt{a^{2}+b^{2}}}$

- wherein

$\rho$ is the distance between $ax+by+c_{1}=0$  and $ax+by+c_{2}=0$

$\frac{x}{5}+ \frac{y}{b}= 1$

Put (13, 32)

$\frac{13}{5}+\frac{y}{b}= 1$

$\Rightarrow b= -20$

$\frac{x}{5}- \frac{y}{20}= 1$

$y = 4x-20$

So, m = 4

Also, $\frac{x}{c}+ \frac{y}{3} = 1\:,\:m = \frac{-3}{c}$

$\frac{-3}{c}= 4\; \Rightarrow c=\frac{-3}{4}$

Equation is -4x + y = 3

Distance $\frac{\left | 3+20 \right |}{\sqrt{17}}=\frac{23}{\sqrt{17}}$

Option 1)

$\frac{23}{\sqrt{15}}\;$

This option is incorrect.

Option 2)

$\; \; \sqrt{17}\;$

This option is incorrect.

Option 3)

$\; \frac{17}{\sqrt{15}}\;$

This option is incorrect.

Option 4)

$\; \frac{23}{\sqrt{17}}$

This option is correct.

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