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Question

# The line 3x+2y=24 meets the y- axis at A and the x -axis at B. The perpendicular bisectors of AB meets the line through (0,−1) parallel to x - axis at C. Find the area of the triangle ABC ?

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Solution

## Given the line, 3x+2y=24⟶(1) for A putting x=0 in (1), 2y=24 ⇒y=12∴ coordinates of A=(0,12) for B putting y=0 in (1), 3y=24 ⇒x=8∴ coordinates of B=(8,0)Midpoint of AB=(8+02,0+122)=(4,6)Now, equation of line perpendicular to line (1), 2x−3y=λIt will pass through (4,6) so, 2(4)−3(6)=λ ⇒λ=8−18 ⇒λ=−10Equation of the line parallel to X-axis is, y=constantit will pass through (0,−1) ⇒−1=constant ⇒y=−1To get coordinates of C,putting y=−1 in 2x−3y=−10 ⇒2x+3=−10 ⇒x=−132Hence, we have C(−132,−1),A(0,12)andB(8,0) ∴ Area of \$\triangle ABC =|−87−4| =|−91| =91squareunits

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