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Question

If the vertices of a triangles are A(7,−1),B(−2,8) and C(1,2) then which of following is/are correct?

A
Altitude through A is x2y9=0
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B
Altitude through B is 2xy+12=0
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C
Altitude through C is xy+1=0
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D
Orthocentre is (2,4)
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Solution

The correct options are
A Altitude through A is x2y9=0
B Altitude through B is 2xy+12=0
C Altitude through C is xy+1=0

Let AD,BE and CF be three altitudes of triangle ABC.
Let m1,m2 and m3 be the slopes of AD,BE and CF respectively.
Then, ADBC
Slope of AD × Slope of BC=1
m1×(281+2)=1
m1=12
BEAC
Slope of BE × Slope of AC=1
m2×(2+117)=1
m2=2
And. CFAB
Slope of CF × Slope of AB=1
m3×(8+127)=1
m3=1
Since AD passes through A(7,1) and has slope of
m1=12
So, its equation is :
y+1=12(x7)
x2y9=0
Similarly, equation of BE is :
y8=2(x+2)
2xy+12=0
Equation of CF is :
y2=1(x1)
xy+1=0

Orthocentre is the intersection of the altitudes.
x2y9=0...(i)
2xy+12=0...(ii)
Solving (i) and (ii) we get
x=11, y=10

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