Find the equations of the tangent and normal to each of the following conics, the lengths of the subtangent and subnormal, then trace the curve showing these lines.
y2 = -9 (x – 3) at (2, 3).
4x2 + 3y2 + 16x + 36y – 68 = 0 at (4, -2)
9x2 – 4y2 – 108x – 56y + 128 = 0 at (0, 2).
x2 + y2 – 8x – 6y + 15 = 0 at (5, 0).
1
Expert's answer
2021-11-25T11:28:54-0500
1.
y2=−9(x−3)
2yy′=−9
y′=2y−9
Point (2,3)
slope=m1=y′=2(3)−9=−23
Tangent line is
y−3=−23(x−2)
The equation of the tangent line in slope-intercept form
y=−23x+6
Find the equation of the normal at the point (2,3)
slope=m2=−m11=32
y−3=32(x−2)
The equation of the normal line in slope-intercept form
y=32x+35
TA is defined to be the subtangent at P. AN is called the subnormal.
The lengths PT and PN are called the tangent and normal.
A(2,0),T(4,0),N(−25,0)
PT=(4−2)2+(0−3)2=13
PN=(−25−2)2+(0−3)2=2313
TA=2
AN=29
2.
4x2+3y2+16x+36y−68=0
8x+6yy′+16+36y′=0
y′=−3y+184x+8
Point (4,−2)
slope=m1=−3(−2)+184(4)+8=−2
Tangent line is
y−(−2)=−2(x−4)
The equation of the tangent line in slope-intercept form
y=−2x+6
Find the equation of the normal at the point (4,−2)
slope=m2=−m11=21
y−(−2)=21(x−4)
The equation of the normal line in slope-intercept form
y=21x−4
TA is defined to be the subtangent at P. AN is called the subnormal.
The lengths PT and PN are called the tangent and normal.
A(4,0),T(3,0),N(8,0)
PT=(3−4)2+(0−(−2))2=5
PN=(8−4)2+(0−(−2))2=25
TA=1
AN=4
3.
9x2−4y2−108x−56y+128=0
18x−8yy′−108−56y′=0
y′=4y+289x−54
Point (0,2)
slope=m1=4(2)+289(0)−54=−23
Tangent line is
y−2=−23(x−0)
The equation of the tangent line in slope-intercept form
y=−23x+2
Find the equation of the normal at the point (0,2)
slope=m2=−m11=32
y−2=32(x−0)
The equation of the normal line in slope-intercept form
y=32x+2
TA is defined to be the subtangent at P. AN is called the subnormal.
The lengths PT and PN are called the tangent and normal.
A(0,0),T(34,0),N(−3,0)
PT=(34−0)2+(0−2)2=3213
PN=(−3−0)2+(0−2)2=13
TA=34
AN=3
4.
x2+y2−8x−6y+15=0
2x+2yy′−8−6y′=0
y′=3−yx−4
Point (5,0)
slope=m1=3−05−4=31
Tangent line is
y−0=31(x−5)
The equation of the tangent line in slope-intercept form
y=31x−35
Find the equation of the normal at the point (5,0)
slope=m2=−m11=−3
y−0=−3(x−5)
The equation of the normal line in slope-intercept form
y=−3x+15
TA is defined to be the subtangent at P. AN is called the subnormal.
The lengths PT and PN are called the tangent and normal.
Comments
Leave a comment