Which of the following statements is true for | vedclass

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(D) Given circle equation: $x^2 + y^2 + 4x - 7y + 12 = 0$.$(1)$ Length of the tangent from point $(x_1, y_1)$ is $\sqrt{x_1^2 + y_1^2 + 4x_1 - 7y_1 +

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None of these.

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(D) Given circle equation: $x^2 + y^2 + 4x - 7y + 12 = 0$.$(1)$ Length of the tangent from point $(x_1, y_1)$ is $\sqrt{x_1^2 + y_1^2 + 4x_1 - 7y_1 + 12}$.For $(1, 2)$: $\sqrt{1^2 + 2^2 + 4(1) - 7(2) + 12} = \sqrt{1 + 4 + 4 - 14 + 12} = \sqrt{7}$.So,option $A$ is incorrect.$(2)$ For $y$-intercept,put $x = 0$: $y^2 - 7y + 12 = 0 \implies (y - 3)(y - 4) = 0$.The roots are $y = 3$ and $y = 4$. The intercept length is $|4 - 3| = 1 
eq 2$.So,option $B$ is incorrect.$(3)$ For $x$-intercept,put $y = 0$: $x^2 + 4x + 12 = 0$.The discriminant $D = b^2 - 4ac = 16 - 4(1)(12) = 16 - 48 = -32 Since $D So,option $C$ is incorrect.$(4)$ Therefore,the correct option is $D$.

Which of the following statements is true for the circle $x^2 + y^2 + 4x - 7y + 12 = 0$?

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