Find whether line is tangent, secant non-intersecting, line equation , $y - 3 x + 2$ circle is equation $x^{2} + y^{2} - 6 x + 5 y - 3 = 0$

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Solution

Line eqn $y = 3 x - 2$
Circle eqn $x^{2} + y^{2} - 6 x + 5 y - 3 = 0$
$⟹ x^{2} + (3 x - 2)^{2} - 6 x + 5 (3 x - 2) - 3 = 0$
$⟹ x^{2} + 9 x^{2} - 12 x + 4 - 6 x + 15 x - 30 - 3 = 0$
$⟹ 10 x^{2} - 3 x - 29 = 0$
$x = \frac{3 \pm \sqrt{9 + 4 * 29 * 10}}{20}$
$x = \frac{3 \pm \sqrt{1169}}{20}$
As the roots are real and there are two solutios hence the line is a secant to the circle

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