Two points P and Q lie on line y=2x+3. These two points P and Q are at a distance 2 units from another point R(1,5). What are the coordinates of the points P and Q?

  1. A. (1+\frac{2}{\sqrt{5}},5+\frac{4}{\sqrt{5}}),(1-\frac{2}{\sqrt{5}},5-\frac{4}{\sqrt{5}}) ✓
  2. B. (3+\frac{2}{\sqrt{5}},5+\frac{4}{\sqrt{5}}),(-1-\frac{2}{\sqrt{5}},5-\frac{4}{\sqrt{5}})
  3. C. (1-\frac{2}{\sqrt{5}},5+\frac{4}{\sqrt{5}}),(1+\frac{2}{\sqrt{5}},5-\frac{4}{\sqrt{5}})
  4. D. (3-\frac{2}{\sqrt{5}},5+\frac{4}{\sqrt{5}}),(-1+\frac{2}{\sqrt{5}},5-\frac{4}{\sqrt{5}})

Correct Answer: A. (1+\frac{2}{\sqrt{5}},5+\frac{4}{\sqrt{5}}),(1-\frac{2}{\sqrt{5}},5-\frac{4}{\sqrt{5}})

Explanation

Point R(1,5) lies on the line y=2x+3 because 2(1)+3=5. P and Q lie on this same line at a distance r=2 from R. The line has slope \tan\alpha = 2, which gives \sin\alpha = \frac{2}{\sqrt{5}} and \cos\alpha = \frac{1}{\sqrt{5}}. Using the parametric form of a line x = x_1 \pm r\cos\alpha and y = y_1 \pm r\sin\alpha, the coordinates are x = 1 \pm 2(\frac{1}{\sqrt{5}}) and y = 5 \pm 2(\frac{2}{\sqrt{5}}). Thus, the points are (1+\frac{2}{\sqrt{5}}, 5+\frac{4}{\sqrt{5}}) and (1-\frac{2}{\sqrt{5}}, 5-\frac{4}{\sqrt{5}}).

Related questions on Analytical Geometry (2D)

Practice more NDA Mathematics questions