coordianteｇｅｏｍｅｔｒｙ(下) 20分

1.
The coordinate of A is (a,0), since A is on the x-axis.

AB is perpendicular to BC.
(slope of AB) x (slope of BC) = -1
(0-8)/(a-4) x (8-6)/(4-0) = -1
4(a-4) = 16
Hence, a = 8

Ans: The coordinate of A is (8,0)

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2.
AB = √[(8-4)2 + (0-8)2]
AB = √(16 + 64)
AB = √80
AB = 4√5 units

BC = √[(4-0)2 + (8-6)2]
BC = √(16 + 4)
AB = 2√5 units

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3.
Area of OABC = Area of ∆ABC + Area of ∆AOC
Area of OABC = (1/2)xABxBC + (1/2)xAOxCO
Area of OABC = (1/2)x(4√5)x(2√5) + (1/2)x8x6
Area of OABC = 20 + 24
Area of OABC = 44 sq.units

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Prove the st. line passing through A(2,3) and B(-4,-6) passes through the origin

The two-point form of the st. line passing through A and B is:
(y-3)/(x-2) = (-6-3)/(-4-2)
(y-3)/(x-2) = 3/2
2(y-3) = 3(x-2)
2y-6 = 3x-6
3x-2y=0

The line is (L): 3x - 2y = 0

Substitute (0,0) into (L):
L.S. = 3(0) - 2(0)
L.S. = 0
L.S. = R.S.

Hence, the origin lies on the st. line.

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How many st. lines are there on the Cartesian coordinate plane with slope 1?

The point slope form of a st. line:
y = mx + c

Slope slope = 1
Hence, the formula of the st. line is
y = x + c

c is any real number.
There are ∞ real numbers.
Therefore, number of st. lines = ∞
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