Conic Sections
Conic Section
star_batch_jee_advanced_2025
Grade 11
Question:
Variable circle is described to pass through point $(1, 0)$ and tangent to the curve $y = \tan(\tan^{-1} x)$. The locus of the centre of the circle is a parabola whose:
length of the latus rectum is $2\sqrt{2}$
axis of symmetry has the equation $x + y = 1$
vertex has the co-ordinates $(\frac{3}{4}, \frac{1}{4})$
length of the latus rectum is $\sqrt{2}$
Step-by-Step Solution
Key Concept: Chord length is found using $2\sqrt{r^2 - d^2}$ where $r$ is radius and $d$ is perpendicular distance from center to chord.
The problem involves finding the length of a chord. Using the distance formula between points $(1/2, 1/2)$ and $(3/4, 1/4)$, and then applying the chord length formula $4a = 2\left|\frac{10-1}{\sqrt{2}}\right| = \frac{2}{\sqrt{2}} = \sqrt{2}$. The calculation uses the perpendicular distance from center to chord and the radius to determine the chord length.
Correct Answer: 2,3,4