if the image of the point(2,1) with respect to a line mirror be (5,2), find the equation of the mirror - Maths - Straight Lines

Question

if the image of the point(2,1) with respect to a line mirror be
(5,2), find the equation of the mirror

Manbar Singh · Accepted Answer

Let M 5, 2 be the image of point L 2, 1 through the mirror PQ, then N be the mid pointof line segment LM & PQ ⊥ LM 
Let h, k be the coordinates of point N then, h = x1 + x22 ; k = y1 + y22h = 2 + 52 ; k = 1 + 22h = 72; k = 32Hence, the coordinates of point N are 72, 32Now, slope of LM = y2 - y1x2 - x1 = 2 - 15 - 2 = 13Since PQ ⊥ LM then slope of PQ = -3 ⇒ m = -3Point N h, k also lies on mirror PQ then using point-slope formula, the equation of the mirror  PQ is:y - k = m x - hy - 32 = -3 x - 72

⇒ 2y - 32= -3 2x - 72⇒ 2y - 3 = -3 2x - 7⇒ 2y - 3 = -6x + 21⇒ 6x + 2y = 21 + 3⇒ 6x + 2y = 24⇒ 2 3x + y = 24⇒3x + y = 12Hence, the equation of mirror PQ is 3x + y = 12

if the image of the point(2,1) with respect to a line mirror be (5,2), find the equation of the mirror.