Tangents to Parabola and Perpendicular Distance Properties

To determine which of the given options are correct, we analyze the geometry of the parabola $y^2 = 4x$.

1. Identify Key Parameters and Tangent Points

For the parabola $y^2 = 4x$, we have $a = 1$. The focus is $S = (1, 0)$.

The equation of a tangent to $y^2 = 4x$ at a parameter $t$ is given by: $yt = x + t^2$

Since the tangents are drawn from the point $P = (-2, 1)$, we substitute the coordinates of $P$ into the tangent equation: $1 \cdot t = -2 + t^2 \implies t^2 - t - 2 = 0 \implies (t - 2)(t + 1) = 0$

Thus, the parameter values for the two points of contact are $t_1 = 2$ and $t_2 = -1$.

• For $t_1 = 2$: $P_1 = (t_1^2, 2t_1) = (4, 4)$
• For $t_2 = -1$: $P_2 = (t_2^2, 2t_2) = (1, -2)$

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2. Analysis for Point $P_1$ and Line $SP_1$

The focus is $S = (1, 0)$ and $P_1 = (4, 4)$. The slope of line $SP_1$ is: $m_1 = \frac{4 - 0}{4 - 1} = \frac{4}{3}$

The equation of line $SP_1$ is: $y - 0 = \frac{4}{3}(x - 1) \implies 4x - 3y - 4 = 0$

$Q_1$ is the foot of the perpendicular from $P(-2, 1)$ onto the line $SP_1$.

Length of $PQ_1$:

Using the perpendicular distance formula: $PQ_1 = \frac{|4(-2) - 3(1) - 4|}{\sqrt{4^2 + (-3)^2}} = \frac{|-8 - 3 - 4|}{5} = \frac{15}{5} = 3$ Thus, Option C is TRUE.

Coordinates of $Q_1$ and Length of $SQ_1$:

Using the foot of perpendicular formula for $P(-2, 1)$ onto $4x - 3y - 4 = 0$: $\frac{x_1 - (-2)}{4} = \frac{y_1 - 1}{-3} = -\frac{4(-2) - 3(1) - 4}{4^2 + (-3)^2} = \frac{15}{25} = \frac{3}{5}$ Solving for $x_1$ and $y_1$: $x_1 = -2 + 4 \left(\frac{3}{5}\right) = \frac{2}{5}$ $y_1 = 1 - 3 \left(\frac{3}{5}\right) = -\frac{4}{5}$ So, $Q_1 = \left(\frac{2}{5}, -\frac{4}{5}\right)$.

Now, the distance $SQ_1$ from focus $S(1, 0)$ is: $SQ_1 = \sqrt{\left(1 - \frac{2}{5}\right)^2 + \left(0 - \left(-\frac{4}{5}\right)\right)^2} = \sqrt{\left(\frac{3}{5}\right)^2 + \left(\frac{4}{5}\right)^2} = \sqrt{\frac{25}{25}} = 1$ Thus, $SQ_1 = 1$, which makes Option A FALSE.

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3. Analysis for Point $P_2$ and Line $SP_2$

The focus is $S = (1, 0)$ and $P_2 = (1, -2)$. Since both $S$ and $P_2$ have $x = 1$, the line $SP_2$ is the vertical line $x = 1$.

$Q_2$ is the foot of the perpendicular from $P(-2, 1)$ onto the line $x = 1$. Thus, $Q_2 = (1, 1)$.

Length of $SQ_2$:

The distance between $S(1, 0)$ and $Q_2(1, 1)$ is: $SQ_2 = \sqrt{(1 - 1)^2 + (1 - 0)^2} = 1$ Thus, Option D is TRUE.

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4. Calculation of Distance $Q_1 Q_2$

Using the points $Q_1 = \left(\frac{2}{5}, -\frac{4}{5}\right)$ and $Q_2 = (1, 1)$: $Q_1 Q_2 = \sqrt{\left(1 - \frac{2}{5}\right)^2 + \left(1 - \left(-\frac{4}{5}\right)\right)^2} = \sqrt{\left(\frac{3}{5}\right)^2 + \left(\frac{9}{5}\right)^2}$ $Q_1 Q_2 = \sqrt{\frac{9 + 81}{25}} = \sqrt{\frac{90}{25}} = \frac{3\sqrt{10}}{5}$ Thus, Option B is TRUE.

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Conclusion

The correct statements are B, C, and D.