This geometry parabola tangents test answer key gives the full solution and the scoring guide for each of the 9 test questions on parabolas from focus and directrix, and tangent lines to a circle.
This answer key gives the worked solution and a scoring guide for each question of the parabola and tangent test. Award points as described, give partial credit for a correct setup, and total the score out of 20 points.
Geometry Parabola Tangents Test Answer Key: questions 1–5
Question 1: Equation from a focus and a directrix (2 points)
(a) \(p=5\) and the vertex is the origin: \(x^{2}=20y\).
(b) \(PF=10\) and the distance to the directrix is \(5+5=10\); also \(10^{2}=20\cdot 5\).
Scoring: 1 point for \(x^{2}=20y\); 1 point for equal distances (or substitution) with the numbers shown.
Question 2: Focus and directrix from an equation (2 points)
\(4p=24\), so \(p=6\). Focus \((0,6)\); directrix \(y=-6\).
Scoring: 1 point for the focus, 1 point for the directrix.
Question 3: A sideways parabola (2 points)
The vertex is \((0,0)\) and \(p=-3\), so \(y^{2}=4(-3)x=-12x\). Check \((-3,6)\): distance to the focus 6, to the directrix \(|-3-3|=6\), and \(36=-12(-3)\).
Scoring: 1 point for the vertex and \(p=-3\); 1 point for \(y^{2}=-12x\).
Question 4: Parabola with a shifted vertex (3 points)
(a) The vertex is the midpoint of \((1,7)\) and \((1,3)\): \((1,5)\), and \(p=2\). The equation is \((x-1)^{2}=8(y-5)\).
(b) For \(x=5\): \(16=8(y-5)\), so \(y=7\).
Scoring: (a) 1 point for the vertex and p, 1 point for the equation; (b) 1 point for \(y=7\).
Question 5: Length of a tangent (2 points)
\(OT\perp PT\), so \(PT=\sqrt{15^{2}-9^{2}}=\sqrt{144}=12\).
Scoring: 1 point for identifying the right angle at T (or writing \(9^{2}+PT^{2}=15^{2}\)); 1 point for 12.
Geometry Parabola Tangents Test Answer Key: questions 6–9
Question 6: Two tangent segments (2 points)
\(2x+7=4x-5\), so \(12=2x\) and \(x=6\). Then \(PA=2(6)+7=19\) (and \(PB=4(6)-5=19\)).
Scoring: 1 point for the equation from congruent tangent segments and \(x=6\); 1 point for \(PA=19\).
Question 7: Constructing a tangent (3 points)
(a) Draw \(OP\); construct its midpoint \(M\); draw the circle with center \(M\) and radius \(MO\); it meets the given circle at \(A\) and \(B\); draw lines \(PA\) and \(PB\).
(b) \(\angle OAP\) is inscribed in a semicircle with diameter \(OP\), so it is \(90^{\circ}\). A line perpendicular to a radius at its end on the circle is tangent.
Scoring: (a) 1 point for the midpoint and the circle centered at M through O, 1 point for the intersection points and the two lines; (b) 1 point for the semicircle (right angle) reason.
Question 8: Angles formed by two tangents (2 points)
(a) Quadrilateral OAPB has right angles at A and B: \(m\angle AOB=360^{\circ}-90^{\circ}-90^{\circ}-48^{\circ}=132^{\circ}\).
(b) OP bisects \(\angle APB\), so \(m\angle OPA=48^{\circ}: 2=24^{\circ}\).
Scoring: 1 point for each part.
Question 9: A satellite dish (2 points)
(a) \(p=2\), so \(x^{2}=8y\).
(b) At the rim \(x=5\): \(y=\frac{25}{8}=3.125\) ft.
Scoring: 1 point per part.
Scoring summary
20 points: add the points of every question of this geometry parabola tangents test answer key.
How to use this geometry parabola tangents test answer key
Use this geometry parabola tangents test answer key to score the test fairly and to see which skills need more practice.
See every chapter on the Geometry math page.