The equation of the lines through the point  $\left(2,3\right)$and making an intercept of length 2 units between the lines $y+2x=3\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}y+2x=5$  is

detailed solution

Correct option is C

As the lines are parallel, distance between them =|5−3|4+1=23=BC∴AB=4−45=45 From △ABC,tan⁡θ=2545=12Slope of the given line is 2. Let m1 be the slope of the required line  Now tan⁡θ=12=m1+21−2m1⇒m1+21−2m1=±12 Consider 12=m1+21−2m1⇒1−2m1=2m1+4⇒4m1=−3⇒m1=−34 Similarly by taking −12 , other value of m1 is undefined.  The equation of the line having slope m and Passing through the point x1,y1 is y−y1=mx−x1⇒y−3=−34(x−2) and y−3=10(x−2)∴ Equations to the required lines are 3x+4y−18=0 and x=2