Question:
If the line
$$
4x - 3y + 7 = 0
$$
touches the circle
$$
x^2 + y^2 - 6x + 4y - 12 = 0
$$
at $ (\alpha, \beta) $, then find $ \alpha + 2\beta $.
If the line
$$
4x - 3y + 7 = 0
$$
touches the circle
$$
x^2 + y^2 - 6x + 4y - 12 = 0
$$
at $ (\alpha, \beta) $, then find $ \alpha + 2\beta $.
Show Hint
Use tangent properties and simultaneous equations to find coordinates.
Updated On: Sep 23, 2025
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The Correct Option is C
Solution and Explanation
1. Tangent condition: distance from center \( (3, -2) \) to line equals radius. 2. Point of contact \( (\alpha, \beta) \) lies on line and circle. 3. Substitute and solve system to get \( \alpha + 2\beta = 1 \).
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