AP EAPCET 24-Aug-2021 Shift 1 Paper

Lines‌x2+kxy+y2 and x+y−1=0 make the sides of equilateral triangle. (given) ∵x2+kxy+y2=0 represent a pair of straight lines passing through origin.
Let m be the slope of one line whose equation is y=mx
According to the question,
tan‌θ=|

m1−m2

1+m1m2

|θ= angle between lines
⇒tan‌60°=|

m−(−1)

1+m(−1)

| [∵ slope of x+y=1 is −1]
⇒√3=

m−1

1−m

⇒(1−m)2=(m+1)2
⇒2m2−8m+2=0
⇒m2−4m+1=0
(

y

x

)2−4(

y

x

)+1=0 [∵y=mx⇒

y

x

=m]
⇒

y2

x2

−

4y

x

+1=0
⇒y2−4xy+x2=0
Compare with x2+kxy+y2=0, we get k=−4
∴k2=(−4)2=16