JEE Main 2025 Apr 2 Shift 1, Mathematics Q25: Involving Two or More Circles

JEE Main2025Apr 2, Shift 1Mathematics

The absolute difference between the squares of the radii of the two circles passing through the point $(- 9, 4)$ and touching the lines $x + y = 3$ and $x - y = 3$ , is equal to _____.

Your answer

Solution

Centre $= (a, 0)$

$r = \frac{a - 0 - 3}{2} = \frac{a - 3}{2}$

Equation of the circle:

$(x - a)^{2} + y^{2} = (\frac{a - 3}{2})^{2}$

Since the circle passes through $(- 9, 4)$ ,

$(- 9 - a)^{2} + 4^{2} = (\frac{a - 3}{2})^{2}$

$2 (a^{2} + 18 a + 81 + 16) = a^{2} - 6 a + 9$

$a^{2} + 42 a + 185 = 0$

$(a + 37) (a + 5) = 0$

$\Rightarrow a = - 37, - 5$

$r_{1} = \frac{- 37 - 3}{2} = 202$

$r_{2} = \frac{- 5 - 3}{2} = 42$

$∣ r_{1}^{2} - r_{2}^{2} ∣ = ∣800 - 32∣ = 768$

$768$

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JEE Main 2025 Apr 2 Shift 1, Mathematics Q25: Involving Two or More Circles

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