JEE Main 2019
Conic Sections
Parabola
Medium
Question
The area (in sq. units) of the smaller of the two circles that touch the parabola, y 2 = 4x at the point (1, 2) and the x-axis is :-
Options
Solution
This problem combines concepts from parabolas and circles, specifically focusing on tangency conditions. We need to find the area of a circle that touches a given parabola at a specific point and also touches the x-axis. The problem asks for the smaller of two such possible circles.
Our strategy will involve:
- Finding the normal to the parabola at the given point, as the center of the circle must lie on this normal.
- Using the condition that the circle touches the x-axis to relate its radius to the y-coordinate of its center.
- Combining these conditions to express the circle's center in terms of its radius.
- Formulating an equation using the distance from the circle's center to the point of tangency