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 __________. [2025]

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Solution

Q.
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 __________. [2025]

Ans.
(768)

We have, x + y = 3 and x – y = 3 are tangents

$∴$ The centre of both circles will lie on x-axis

Equation of circle is

$∴ {(x - α)}^{2} + y^{2} = r^{2}$

Hence, centre is C( $α$ , 0).

$r = P C = \sqrt{{(α + 9)}^{2} + 16}$ ... (i)

Also, $| \frac{α - 3}{\sqrt{2}} |$ $= r$ ... (ii)

From (i) and (ii), we get

$\sqrt{{(α + 9)}^{2} + 16} =$ $| \frac{α - 3}{\sqrt{2}} |$

$\Rightarrow α = - 5 or - 37$

$r =$ $| \frac{- 5 - 3}{\sqrt{2}} |$ $or$ $| \frac{- 37 - 3}{\sqrt{2}} |$ $= 4 \sqrt{2} or 20 \sqrt{2}$

$∴$ Now, $| r_{1}^{2} - r_{2}^{2} |$ $=$ $| {(4 \sqrt{2})}^{2} - {(20 \sqrt{2})}^{2} |$ $= 768$ .

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