গণিত প্রশ্ন

AI-তৈরি নমুনা উত্তর

The equation of the circle is:
$x^2 + y^2 + px + qy + 2 = 0$

Since the circle passes through $(0, -2)$:
$0^2 + (-2)^2 + p(0) + q(-2) + 2 = 0$
$4 - 2q + 2 = 0 \implies 2q = 6 \implies q = 3$

Since the circle passes through $(-2, 0)$:
$(-2)^2 + 0^2 + p(-2) + q(0) + 2 = 0$
$4 - 2p + 2 = 0 \implies 2p = 6 \implies p = 3$

The center of the circle is given by:
$\left(-\frac{p}{2}, -\frac{q}{2}\right) = \left(-\frac{3}{2}, -\frac{3}{2}\right)$

Since the center lies on the line $ax - y = 3$:
$a\left(-\frac{3}{2}\right) - \left(-\frac{3}{2}\right) = 3$
$-\frac{3}{2}a + \frac{3}{2} = 3$
$-\frac{3}{2}a = \frac{3}{2}$
$a = -1$

Thus, the values are:
$a = -1, \quad p = 3, \quad q = 3$