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Conic Sections

Parabolas, ellipses, and hyperbolas, oh my! Learn about this eccentric bunch of shapes.

Parabola - Focus and Directrix

         

The above diagram shows a point \(P\) on a parabola \(y^2=12x\) with focus \(F\). If \(l\) is a tangent line to the parabola at point \(P\), the \(x\)-intercept of \(l\) is \(Q\), and \(\angle FPQ=36^\circ,\) what is \(\angle PFQ\) in degrees?

Note: The above figure is not drawn to scale.

The above diagram shows a parabola with vertex on the \(y\)-axis, and directrix parallel to the \(y\)-axis. If the focus of the parabola is \(F=(4,6)\), what is the equation for the parabola?

Note: The above diagram is not drawn to scale.

If \(y=ax+b\) is the equation for a line with slope \(6\) that is a tangent line of the parabola \(y^2=12x\), what is \(a+b?\)

Identify the focus and directrix of the parabola given by \(x =-\frac{1}{28}y^{2}.\)

In the diagram above, points \(P, Q,\) and \(R\) lie on parabola \(y^2=60x.\) If the center of gravity of \(\triangle PQR\) is the focus \(F\) of the parabola, what is the sum of the lengths \(\overline{PF}, \overline{QF},\) and \(\overline{RF}?\)

Note: The figure above is not drawn to scale.

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