Two vertices of a triangle are `(-2, -1)` and `(3, 2)` and third vertex lies on the line `x+y=5.` If the area of the triangle is `8` square units, then the third vertex is:

Options

A. ` ( 0,5 ) or ( 1, 4 ) `
B. ` (5, 0 ) or ( 4, 1 ) `
C. ` ( 5, 0 or ( 1, 4 ) `
D. ` (0, 5 ) or ( 4, 1 ) `

Answer

C

Explanation

Since the third vertex ` ( x _ 1, y _ 1 ) ` lie on the line ` x + y = 5`.

` therefore x _ 1 + y _ 1 = 5 `
` y _ 1 = 5 - x _1 `
` therefore ` Coordinate of C is ` ( x _ 1 , 5 - x _ 1 ) `
Given, area of ` Delta ABC = 4 ` units
` therefore (1 )/ (2 ) |{:( 2 ,, - 1,, 1 ), ( 3,, 2 ,, 1 ),( x _ 1 ,, 5 - x _ 1 ,, 1 ) :}| = 4 `
Using ` R _ 2 to R _ 2 -R _ 1 and R _ 3 to R _ 3 - R _ 1 `,
` |{:( 2,, - 1,, 1 ), ( 1,,3,,0),( x _ 1 - 2,, 6 - x _1 ,, 0 ):}| = 8 `
` rArr 6 - x _1 -3 ( x _ 1 -2 ) = pm 8 `
` rArr 6 - x _ 1 - 3 x _ 1 + 6 = pm 8 `
` rArr 12 - 8 4 x _ 1 or 4x _ 1 = 20 `
` rArr x _ 1 = 1 or x _ 1 = 5 `
` therefore y _ 1 = 5 - 1 = 4 or y_ 1 = 0 `
` therefore C ( x _ 1 , y _ 1 ) =C ( 1, 4 ) or C ( 5, 0 ) `