Question

The circle passing through the distinct points $\left(1,t\right),\left(t,1\right)$ and $\left(t,t\right)$ for all values of ${}^{\prime }{t}^{\prime }$ , passes through the point:

• A. $\left(-1,-1\right)$
• B. $\left(-1,1\right)$
• C. $\left(1,-1\right)$
• D. $\left(1,1\right)$

Answer 1

Answer: (D)

$\left(1,1\right)$

We know that the general equation of the circle lies the point $(h,k)$ are,

${(x-h)}^2+{(y-k)}^2=r^2.......\left(1\right)$

If it passes through the points $(1,t)$.

${(1-h)}^2+{(t-k)}^2=r^2.......\left(2\right)$

If it passes through the points $(t,t)$.

${(t-h)}^2+{(t-k)}^2=r^2.......\left(3\right)$

If it passes through the points $(t,1)$.

${(t-h)}^2+{(1-k)}^2=r^2.......\left(4\right)$

On solving equation (2) and (3) to,

$t-h=1-h\Rightarrow t=1$

If $t=1$, then the circle (1) passes through(1,1)

Hence D is the co0rrect option