Question 38 Let L be the straight line passing through P (-1,-1) with angle of inclination to the x-axis. It is known that the coordinates of any point Q on L are in the form (-1+ r cos 0, -1½-½ + r sin 0), where r is a real number. (a) Show that PQ = \r\ In the figure below, L cuts the parabola y = 3x² + 2 at point A and B. Let PA = r₁ and PB = r₂. y y = 3x² +2 B L P(-1,-) 8 Ο (b) It is known that the points A and B lie both on the line L and the parabola y = 3x² + 2. Show that r₁ and 2 are the roots of the equation 9r2 cos20-3r (sin 0 + 6 cos 0) +16=0 (c) Using (b), show that AB² = (sin 0 - 2 cos 0) (sin 0 + 14 cos 0) 9 cos⭑0 (d) Let L₁ be a tangent to the parabola y = 3x² +2 from P, with point of contact R. Using the results above, find the two possible slopes of L₁. (e) Show that PR = 4√5 when one of the slopes of L₁ has a value of 2.
Question 38 Let L be the straight line passing through P (-1,-1) with angle of inclination to the x-axis. It is known that the coordinates of any point Q on L are in the form (-1+ r cos 0, -1½-½ + r sin 0), where r is a real number. (a) Show that PQ = \r\ In the figure below, L cuts the parabola y = 3x² + 2 at point A and B. Let PA = r₁ and PB = r₂. y y = 3x² +2 B L P(-1,-) 8 Ο (b) It is known that the points A and B lie both on the line L and the parabola y = 3x² + 2. Show that r₁ and 2 are the roots of the equation 9r2 cos20-3r (sin 0 + 6 cos 0) +16=0 (c) Using (b), show that AB² = (sin 0 - 2 cos 0) (sin 0 + 14 cos 0) 9 cos⭑0 (d) Let L₁ be a tangent to the parabola y = 3x² +2 from P, with point of contact R. Using the results above, find the two possible slopes of L₁. (e) Show that PR = 4√5 when one of the slopes of L₁ has a value of 2.
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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Transcribed Image Text:Question 38
Let L be the straight line passing through P (-1,-1) with angle of inclination to the x-axis. It is known that the
coordinates of any point Q on L are in the form (-1+ r cos 0, -1½-½ + r sin 0), where r is a real number.
(a) Show that PQ = \r\
In the figure below, L cuts the parabola y = 3x² + 2 at point A and B. Let PA = r₁ and PB = r₂.
y
y = 3x² +2
B
L
P(-1,-)
8
Ο
(b) It is known that the points A and B lie both on the line L and the parabola y = 3x² + 2.
Show that r₁ and 2 are the roots of the equation 9r2 cos20-3r (sin 0 + 6 cos 0) +16=0
(c) Using (b), show that
AB² =
(sin 0 - 2 cos 0) (sin 0 + 14 cos 0)
9 cos⭑0
(d) Let L₁ be a tangent to the parabola y = 3x² +2 from P, with point of contact R.
Using the results above, find the two possible slopes of L₁.
(e) Show that PR = 4√5 when one of the slopes of L₁ has a value of 2.
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