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NDA 1 2024 Maths, question 60

Topic: Analytical geometry

Q60

Two points PP and QQ lie on line y=2x+3y=2x+3. These two points PP and QQ are at a distance 2 units2\text{ units} from another point R(1,5)R(1,5). What are the coordinates of the points PP and QQ ?

  1. (a) (1+25,5+45),(1−25,5−45)\left( 1+\frac{2}{\sqrt{5}},5+\frac{4}{\sqrt{5}} \right),\left( 1-\frac{2}{\sqrt{5}},5-\frac{4}{\sqrt{5}} \right)Correct answer
  2. (b) (3+25,5+45),(−1−25,5−45)\left( 3+\frac{2}{\sqrt{5}},5+\frac{4}{\sqrt{5}} \right),\left( -1-\frac{2}{\sqrt{5}},5-\frac{4}{\sqrt{5}} \right)
  3. (c) (1−25,5+45),(1+25,5−45)\left( 1-\frac{2}{\sqrt{5}},5+\frac{4}{\sqrt{5}} \right),\left( 1+\frac{2}{\sqrt{5}},5-\frac{4}{\sqrt{5}} \right)
  4. (d) (3−25,5+45),(−1+25,5−45)\left( 3-\frac{2}{\sqrt{5}},5+\frac{4}{\sqrt{5}} \right),\left( -1+\frac{2}{\sqrt{5}},5-\frac{4}{\sqrt{5}} \right)

Solution.

Answer: (a) (1+25,5+45),(1−25,5−45)\left( 1+\frac{2}{\sqrt{5}},5+\frac{4}{\sqrt{5}} \right),\left( 1-\frac{2}{\sqrt{5}},5-\frac{4}{\sqrt{5}} \right)

  1. The line has unit direction vector (1, 2)/√5.
  2. Add and subtract twice this vector from R = (1, 5).
  3. The points are (1 + 2/√5, 5 + 4/√5) and (1 − 2/√5, 5 − 4/√5), option (a).
  4. Each lies on the line and its squared distance from R is 4.