গণিত প্রশ্ন

AI-তৈরি নমুনা উত্তর

Given the equation of the circle:
$x^2 + y^2 - 8x - 2y + 4 = 0$

Comparing with the standard circle equation $x^2 + y^2 + 2gx + 2fy + c = 0$:
Center, $(h, k_c) = (4, 1)$
Radius, $r = \sqrt{4^2 + 1^2 - 4} = \sqrt{13}$

The given line is:
$3x + ky - 1 = 0$

Since the line touches the circle, the perpendicular distance from the center $(4, 1)$ to the line is equal to the radius $r$:
$\frac{|3(4) + k(1) - 1|}{\sqrt{3^2 + k^2}} = \sqrt{13}$

$\Rightarrow \frac{|11 + k|}{\sqrt{9 + k^2}} = \sqrt{13}$

Squaring both sides:
$\frac{(11 + k)^2}{9 + k^2} = 13$
$\Rightarrow 121 + 22k + k^2 = 13(9 + k^2)$
$\Rightarrow 121 + 22k + k^2 = 117 + 13k^2$
$\Rightarrow 12k^2 - 22k - 4 = 0$
$\Rightarrow 6k^2 - 11k - 2 = 0$
$\Rightarrow 6k^2 - 12k + k - 2 = 0$
$\Rightarrow 6k(k - 2) + 1(k - 2) = 0$
$\Rightarrow (k - 2)(6k + 1) = 0$

$\therefore k = 2$ or $k = -\frac{1}{6}$