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Question

Let k be an integer such that the triangle with vertices (k,3k),(5,k) and (k,2) has area 28 sq. units. Then the orthocentre of this triangle is at the point

A
(1,34)
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B
(1,34)
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C
(2,12)
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D
(2,12)
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Solution

The correct option is D (2,12)
Given vertices of triangle are (k,3k),(5,k)and(k,2)
Area of triangle =28
12|k(k2)5(23k)k(3kk)|=28

k22k+10+15k+4k2=±565k2+13k46=0and5k2+13k+66=0
k=2
the cordinates of vertices of triangle are A(2,-6),B(5,2) and C(-2,2)

Now ,Equation of altitude from vertex A,
y(6)=12225(x2)

x=2 (1)

Now equation of alitude from vertex c is
y2=16225(x(2))
3x+8y10=0 (2)
On solving 1 and 2 we get x=2 and y=12

Therefore Orhocenter is (2,12)

1909108_1220998_ans_73f1514ad3fd47a6972ee78ffa7768b1.png

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