Let A a , b , B 3 , 4 and − 6 , − 8 respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point P 2 a + 3 , 7 b + 5 from the line 2 x + 3 y − 4 = 0 measured parallel to the line x − 2 y − 1 = 0 is

Step-by-step solution

Given: A a , b , B 3 , 4 , C − 6 , − 8 denotes centroid, circumcentre and orthocentre respectively. We know that, the line joining orthocentre and circumcentre is divided by centroid in the ratio 2 : 1 , where distance from orthocentre being the larger part. ⇒ a = 2 × 3 + 1 - 6 2 + 1 , b = 2 × 4 + 1 - 8 2 + 1 ⇒ a = 0 , b = 0 ⇒ P ≡ 2 × 0 + 3 , 7 × 0 + 5 ⇒ P ≡ 3 , 5 Distance from P measured along x − 2 y − 1 = 0 ⇒ x = 3 + r cos θ , y = 5 + r sin θ Where, tan θ = 1 2 So, the distance of point from 2 x + 3 y − 4 = 0 wi