Questions
Let be an integer such that the triangle with vertices has area 28 sq. units. Then the orthocentre of this triangle is at the point:
detailed solution
Correct option is D
Let ABC be the triangle the coordinates of whose vertices are ;4 (k, -3k), B, (5, k) and C(-k, 2). It is given that Area of∆ABC = 28 sq. units =12k −3k 15 k 1−k 2 1=±28⇒ 12k −3k 15−k 4k 0−2k 2+3k 0=±28=(5−k)(2+3k)+8k2=±56=5k2+13k+66=0 or 5k2+13k−46=0⇒5k2+13k−46=0 ∵5k2+13k+66=0 has imaginary roots ⇒(k−2)(5k+23)=0⇒k=2 [∵k∈Z]Hence, the coordinates of vertices are A ( 2, - 6), B (5, 2) and(−2, 2)The equations of altitudes through vertices A. andC are .:x= 2 and 3x + By -10 = 0 respectively. These two altitudes intersect : (2, 1/2). Hence, the coordinates of the orthocentre are ( 2, 1/ 2).Talk to our academic expert!
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