Question

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

• A. y=2x+1
• B. y=-2x+1
• C. y-2x+1 = 0
• D. 2y-x+1 = 0

Answer 1

Answer: (A), (B)

The correct options are
A

y=2x+1

B

y=-2x+1

$\underset{–}{FindingB}$
It is the intersection of y=x and y=k
$\Rightarrow$ (k,k) is the point of intersection.

$\underset{–}{FindingA}$
It is the intersection of y=x and x+y=2
$\Rightarrow$ (1,1) is A.

$\underset{–}{FindingC}$
C is the intersection of y=x and x+y=2
$\Rightarrow$ (2-k,k) is C.

Area of $△=\frac{1}{2}$ $AD\times BC$
BC=2-k-k = 2-2k
AD=k-1
Area of $△=\frac{1}{2}2(1-k)(k-1)$
=$(k-1{)}^2$(Area is +ve)
This is given as $4h^2$
$\Rightarrow 4h^2=(k-1{)}^2$
$2h=\pm (k-1)$
$\Rightarrow 2h-k+1=0or2h+k-1=0$
Replace(h,k) with (x,y)
$\Rightarrow$ 2x-y+1=0 or 2x+y-1=0
Or y=2x+1 or y=-2x+1