What is the equation of the pair of tangents | vedclass

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(C) The equation of the pair of tangents from a point $(x_1, y_1)$ to a parabola $S = 0$ is given by $SS_1 = T^2$.Here,$S = y^2 - 12x = 0$ and the poi

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$3x^2 + y^2 + 10x - 4xy - 4y + 3 = 0$

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(C) The equation of the pair of tangents from a point $(x_1, y_1)$ to a parabola $S = 0$ is given by $SS_1 = T^2$.Here,$S = y^2 - 12x = 0$ and the point is $(x_1, y_1) = (1, 4)$.$S_1 = (4)^2 - 12(1) = 16 - 12 = 4$.The tangent $T$ at $(x_1, y_1)$ is given by $yy_1 - 6(x + x_1) = 0$,which is $4y - 6(x + 1) = 4y - 6x - 6$.Substituting into $SS_1 = T^2$:$(y^2 - 12x)(4) = (4y - 6x - 6)^2$.$4y^2 - 48x = 16y^2 + 36x^2 + 36 - 48xy - 48y + 72x$.Rearranging the terms:$36x^2 + 12y^2 - 48xy + 120x - 48y + 36 = 0$.Dividing by $12$:$3x^2 + y^2 - 4xy + 10x - 4y + 3 = 0$.

What is the equation of the pair of tangents drawn from the point $(1, 4)$ to the parabola $y^2 = 12x$?

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