Question

Find the points on the line 3x- 4 y-1= 0 which are at a distance of 5 units from the point (3, 2).​

Answer 1

Let P(h,k) is a point on given line 3x−4y−1=0

3h−4k−1=0 or h=3(4k+1) …..(1)

Distance between a point A(2,3) and P(h,k) is 5 units

PA=5

PA2=25

(h−2)2+(k−3)2=25 on putting h=3(4k+1) from equation (1)

{(4k+1)(3−2)2}+(k−3)2=25

(a1)(4k−5)2+(k−3)2=25

(4k−5)2+9(k−3)2=225

16k2−40k+25+9k2−34k+81=225

25k2−94k+106−225=0

25k2−94k−119=0

(25k−119)(k+1)=0

k=25119,−1 on putting the value of k in equation (1)

h=3(4k+1) ……(1)

h=3(25476+1)

=25×3501

=25167

(or) h=3(−4+1)=−1

Required points are (25167,25119) & (−1,−1)