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# Straight Line

Straight Line L passes through point (1,1) and its distance from point (4,5) is 2 units. Find :

(1) Equation(s) of L.

(2) Point on L closest to (4,5).

Note : Using formula outside the DSE syllabus is not allowed.

### 3 Answers

- 50418129Lv 78 years agoFavorite Answer
Let P(x, y) be the projection of (4, 5) on L

since L is perpendicular to the line joined P and (4,5),

[(y - 1)/(x - 1)] [(y - 5)/(x - 4)] = -1

(y - 1)(y - 5) = -(x - 1)(x - 4)

x² + y² - 5x - 6y + 9 = 0 ---------------------------------------------- (i)

since the distance from P to (4, 5) = 2

√[(x - 4)² + (y - 5)²] = 2

(x - 4)² + (y - 5)² = 4

x² + y² - 8x - 10y + 37 = 0 ----------------------------------------- (ii)

(i) - (ii), 3x + 4y - 28 = 0 => x = (28 - 4y)/3

substitute x = (28 - 4y)/3 into (i),

[(28 - 4y)/3]² + y² - 5[(28 - 4y)/3] - 6y + 9 = 0

(28 - 4y)² + 9y² - 15(28 - 4y) - 54y + 81 = 0

16y² - 224y + 784 + 9y² - 420 + 60y - 54y + 81 = 0

25y² - 218y + 445 = 0

y = (218 ± √3024) / 50 = (109 ± 6√21) / 25

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