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The area of the triangle formed by the intersection of a line parallel to xaxis and passing through P(h,k) with the lines y=x and x+y=2 is 4h2. Then the locus of the point P is

A
y=3x2 or y=3x2
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B
y=2x1 or y=2x1
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C
y=3x+2 or y=3x+2
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D
y=2x+1 or y=2x+1
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Solution

The correct option is D y=2x+1 or y=2x+1
A line passing through P(h,k) and parallel to x axis is y=k ........................(1)
The other two given lines are
y=x ........................(2)
and x+y=2 .........................(3)
Solving eqns (1) and (2)
we get A(k,k)
Solving eqns (2) and (3)
we get B(1,1)
Solving eqns (3) and (1)
we get C(2k,k)
Area of ABC=4h2
12∣ ∣kk11112kk1∣ ∣=4h2
Applying C2C2C1, then
12∣ ∣k011012k2k21∣ ∣=4h2
(k1)2=4h2
k1=±2h
k=±2h+1
Replace hx,ky
Locus of (h,k) is y=2x+1 or y=2x+1

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