UPSC NDA Math Model Paper 4

Given :The length of the perpendicular from the point (4, 1) on the line 3x - 4y + k = 0 is 2 units.
Let P = (4, 1)Here x1=4,y1=1,a=3,b=−4 and d = 2
Now substitute x1=4 and y1=1 in the equation 3x−4y+k=0
⇒|3.x1−4.y1+k| =|12−4+k| =|8+k|
⇒√a2+b2=√32+(−4)2=5
As we know that, the perpendicular distance d from P(x1,y1) to the line ax+by+c=0 is given by d=|

ax1+by1+c

√a2+b2

|
⇒d=|

3.x1−4.y1+k

√32+(−4)2

|=2
⇒|8+k|=10
⇒k=2 or -18
Hence, option C is the correct answer.