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Determinants

Short Answer Type

141.

Find values of k if area of triangle is 4 sq. units and vertices are (k, 0), (4, 0), (0, 2).

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142.

Find values of k if area of triangle is 4 sq. units and vertices are (– 2, 0), (0, 4), (0, k)

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143. Find the equation of the line joining A(l, 3) and B(0, 0) using determinants and find k if D(A, 0) is a point such that area of triangle ABD is 3 sq. units.

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144.

Find equation of line joining (1, 2) and (3, 6) using determinants.

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145.
Find equation of line joining (3, 1) and (9, 3) using determinants.

Let P(x, y) be any point on the line joining A(3, 1) and B(9, 3).

therefore space space area space of space triangle space ABP space equals space 0 therefore space space space 1 half open vertical bar table row 3 cell space space space 1 end cell cell space space 1 end cell row 9 cell space space 3 end cell cell space space 1 end cell row straight x cell space space straight y end cell cell space space 1 end cell end table close vertical bar space equals space 0 space space space space space space space space space space space rightwards double arrow space space space space space open vertical bar table row 3 cell space space 1 end cell cell space space 1 end cell row 9 cell space space 3 end cell cell space space 1 end cell row straight x cell space space straight y end cell cell space space 1 end cell end table close vertical bar space equals 0 space therefore space space space space space space space space 3 open vertical bar table row 3 cell space space 1 end cell row straight y cell space space 1 end cell end table close vertical bar minus 1 open vertical bar table row 9 cell space space space 1 end cell row straight x cell space space space 1 end cell end table close vertical bar plus 1 open vertical bar table row 9 cell space space 3 end cell row straight x cell space space space straight y end cell end table close vertical bar space equals space 0 therefore space space space space space space space space space 3 left parenthesis 3 minus straight y right parenthesis minus 1 left parenthesis 9 minus straight x right parenthesis plus 1 left parenthesis 9 straight y minus 3 straight x right parenthesis space equals space 0 therefore space space space space space space space space space 9 minus 3 straight y minus 9 plus straight x plus 9 straight y minus 3 straight x space equals space 0 therefore space space space space space space space space space minus 2 straight x plus 6 straight y space equals space 0 space space space space space space space space space space space space or space space space space straight x minus 3 straight y space equals space 0

which is required equation of line.

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Long Answer Type

146. A triangle has its three sides equal to a, b and c. If the coordinates of its vertices are A(x₁ y₁), B(x₂, y₂) and C(x₃ y₃), show that

space open vertical bar table row cell straight x subscript 1 end cell cell space space straight y subscript 1 end cell cell space 2 end cell row cell straight x subscript 2 end cell cell space straight y subscript 2 end cell cell space 2 end cell row cell straight x subscript 3 end cell cell space straight y subscript 3 end cell cell space 2 end cell end table close vertical bar squared space equals space left parenthesis straight a plus straight b plus straight c right parenthesis thin space left parenthesis straight b plus straight c minus straight a right parenthesis thin space left parenthesis straight c plus straight a minus straight b right parenthesis thin space left parenthesis straight a plus straight b minus straight c right parenthesis.

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Multiple Choice Questions

147. If area of triangle is 35 sq. units with vertices (2, – 6), (5, 4) and (k, 4). Then k is

12
-2
-12, -2
-12, -2

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Short Answer Type

148.

Find the minor of element 6 in the determinant $increment space equals space open vertical bar table row 1 cell space 2 end cell cell space 3 end cell row 4 cell space 5 end cell cell space 6 end cell row 7 cell space 8 end cell cell space 9 end cell end table close vertical bar$

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149.

Find the minors and co-factors of the elements a₁₁, a₂₁ in the determinant.

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150.

Find minors and cofactors of all the elements of the determinant:
$open vertical bar table row 1 cell space space minus 2 end cell row 4 cell space space space 3 end cell end table close vertical bar$ .