Question

Four distinct points $( 2 k , 3 k ) , ( 1,0 ) ( 0,1 )$ and $( 0,0 )$ lie on a circle for.

• A. $\forall k \in I$
• B. $k < 0$
• C. $0 < k < 1$
• D. For two values of k

Answer 1

Answer: (D)

For two values of k

Answer 2

General equation of circle is,

$x ^ { 2 } + y ^ { 2 } + 2 g x + 2 f y + c = 0$

It passes through (0,0), (1, 0) and (0, 1); ∴ $c = 0$

Now $2 g + 1 = 0 \Rightarrow g = - \frac { 1 } { 2 }$ and $2 f + 1 = 0 \Rightarrow f = \frac { - 1 } { 2 }$

Hence equation of circle is $x ^ { 2 } + y ^ { 2 } - x - y = 0$

Point $( 2 k , 3 k )$lies on the circle

∴$4 k ^ { 2 } + 9 k ^ { 2 } - 5 k = 0$

⇒ $13 k ^ { 2 } - 5 k = 0$

⇒ $k = 0$ or $k = \frac { 5 } { 13 }$.