A line through origin meets the line 2x = 3y + 13 at right angles at point Q. Find the co-ordinates of Q.
Straight Line Eq
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Question

A line through origin meets the line 2x = 3y + 13 at right angles at point Q. Find the co-ordinates of Q.

KnowledgeBoat · Accepted Answer

Let the line passing through the origin O(0, 0) be L1 and slope m1, and the given line be L2 : 2x = 3y + 13 and slope be m2. Given equation of line, ⇒ 2x = 3y + 13 ⇒ 3y = 2x - 13 ⇒ y = Comparing above equation with y = mx + c we get, Slope (m2) = . Since L1 is perpendicular to L2, the product of their slopes is -1. ⇒ m1 × m2 = -1 ⇒ m1 × = -1 ⇒ m1 = . The equation of the line L1 having slope m1 and passing through the origin can be given by point-slope form i.e., ⇒ y - y1 = m(x - x1) ⇒ y - 0 = (x - 0) ⇒ y = ⇒ 2y = -3x ⇒ 2y + 3x = 0. For finding the coordinates of the foot of the perpendicular which is the point of intersection of the lines. -3y + 2x - 13 = 0 .......(i) 2y + 3x = 0 ..........(ii) On multiplying equation (i) by 2, we get : -6y + 4x - 26 = 0 ..........(iii) On multiplying equation (ii) by 3 we get, 6y + 9x = 0 .........(iv) Adding (iii) and (iv) we get, ⇒ -6y + 4x - 26 + 6y + 9x = 0 ⇒ 13x - 26 = 0 ⇒ 13x = 26 ⇒ x = ⇒ x = 2. Substituting value of x in (ii), we get : ⇒ 2y + 3(2) = 0 ⇒ 2y + 6 = 0 ⇒ 2y = -6 ⇒ y = ⇒ y = -3. Coordinates = (2, -3) Hence, coordinates of Q = (2, -3).

Mathematics

A line through origin meets the line 2x = 3y + 13 at right angles at point Q. Find the co-ordinates of Q.

Straight Line Eq

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