JEE PYQ: Circle Question 31
Question 31 - 2019 (10 Apr Shift 2)
The locus of the centres of the circles, which touch the circle, $x^2 + y^2 = 1$ externally, also touch the $y$-axis and lie in the first quadrant, is:
(1) $x = \sqrt{1 + 4y}, y \ge 0$
(2) $y = \sqrt{1 + 2x}, x \ge 0$
(3) $y = \sqrt{1 + 4x}, x \ge 0$
(4) $x = \sqrt{1 + 2y}, y \ge 0$
Show Answer
Answer: (2)
Solution
Let center $(h,k)$ with $h > 0, k > 0$. Touches $y$-axis: $r = h$. Touches $x^2+y^2=1$ externally: $\sqrt{h^2+k^2} = 1 + h$. So $k^2 = 1 + 2h$. Locus: $y = \sqrt{1+2x}$.
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