Question

Sol. [Key Concept: Since $tan(\tan^{- 1}x) = x$, a circle through $(1,0)$ tangent to $y = x$ has its centre equidistant from the point and the line — a parabola; read off its latus rectum, axis and vertex.]

Since $y = tan(\tan^{- 1}x) = x$, a circle through $(1,0)$ and tangent to $y = x$ has its centre equidistant from the point $(1,0)$ and the line $x - y = 0$ — a parabola with focus $(1,0)$, directrix $y = x$.

Axis is perpendicular to the directrix through the focus (slope $- 1$): $x + y = 1$ → (b).

Foot of perpendicular from $(1,0)$ on $y = x$ is $(\frac{1}{2},\frac{1}{2})$, so vertex = midpoint $(\frac{3}{4},\frac{1}{4})$ → (c).

Latus rectum $= 2 \times (focus - directrix\ distance) = 2 \cdot \frac{1}{\sqrt{2}} = \sqrt{2} \neq \ 2\sqrt{2}$, so (a) is false.