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99 Percentile Qs Bank for JEE MainMathematicsStraight Lines

A quadrilateral A B C D is divided by the diagonal A C into two triangles of equal areas. If A ,   B ,   C are respectively 3 , 4 ,   - 3 , 6 ,   - 5 , 1 , then the locus of D is

Options

  1. Ax - 8 y - 57 x - 8 y + 11 = 0
  2. Bx - 8 y - 57 x - 8 y - 11 = 0
  3. C3 x - 8 y - 57 3 x - 8 y + 11 = 0
  4. D3 x - 8 y - 11 3 x - 8 y + 57 = 0

Correct answer

D. 3 x - 8 y - 11 3 x - 8 y + 57 = 0

Step-by-step solution

Given points are A ( 3 , 4 ) ,   B ( - 3 , 6 ) ,   C ( - 5 , 1 ) ,   D ( x , y ) . Here area of A B C = Area of A C D . Area of A B C = 1 2 ∑ x 1 y 2 - y 3 = 1 2 3 6 - 1 + - 3 1 - 4 + - 5 4 - 6 = 1 2 3 5 + - 3 - 3 + - 5 - 2 = 1 2 15 + 9 + 10 = 17     . . . 1 Area of A C D = 1 2 ∑ x 1 y 2 - y 3 = 1 2 3 1 - y + - 5 y - 4 + x 4 - 1 = 1 2 3 - 3 y - 5 y + 20 + 3 x = 1 2 + 3 x - 8 y + 23     . . . 2 From equations ( 1 ) and ( 2 ) , ± ( + 3 x - 8 y + 23 ) = 34 ⇒ 3

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