
(a)
To find:
The equation of the set of all points from the curve of slops reciprocal.
(a)

Answer to Problem 20P
β=mα+c
Explanation of Solution
Given:
The equation of ellipse
x2a2+y2b2=1
Where a≠b
Concept used:
A tangent line to the ellipse
y=mx+c
The line intersect exactly one point on the ellipse the discriminant of the equation is zero
b2−4ac=0
Calculation:
The standard form of ellipse equation
x2a2+y2b2=1....................(1)
A tangent line to the ellipse
y=mx+c
The line intersect exactly one point on the ellipse
Substitute y=mx+c in equation (1)
x2a2+y2b2=1x2a2+(mx+c)2b2=1x2a2+(m2x2+c2+2mcx)b2=1b2x2+m2x2a2+a2c2+2mxca2=a2b2(b2+a2m2)+2mca2x+a2c2−a2b2=0
The line intersect exactly one point on the ellipse the discriminant of the equation is zero
b2−4ac=0(2mca2)2−4(b2+a2m2)(a2c2−a2b2)=0−4b2a2c2+4a2b4+4m2a4b2=0
Simplifying the term
4a2b2(−c2+b2+a2m2)=0a2m2−c2+b2=0
The point (α,β) lies on the line y=mx+c
So, β=mα+c
(b)
To find:
The equation of the set of all points from the curve of slops are negative reciprocal.
(b)

Answer to Problem 20P
β=−mα+c
Explanation of Solution
Given:
The equation of ellipse
x2a2+y2b2=1
Where a≠b
Concept used:
A tangent line to the ellipse
y=mx+c
The line intersect exactly one point on the ellipse the discriminant of the equation is zero
b2−4ac=0
Calculation:
The standard form of ellipse equation
x2a2+y2b2=1....................(1)
A tangent line to the ellipse
y=−mx+c
The line intersect exactly one point on the ellipse
Substitute y=mx+c in equation (1)
x2a2+y2b2=1x2a2+(−mx+c)2b2=1x2a2+(m2x2+c2−2mcx)b2=1b2x2+m2x2a2+a2c2−2mxca2=a2b2(b2+a2m2)−2mca2x+a2c2−a2b2=0
The line intersect exactly one point on the ellipse the discriminant of the equation is zero
b2−4ac=0(−2mca2)2−4(b2+a2m2)(a2c2−a2b2)=0−4b2a2c2+4a2b4+4m2a4b2=0
Simplifying the term
4a2b2(−c2+b2+a2m2)=0a2m2−c2+b2=0
The point (α,β) lies on the line y=−mx+c
So, β=−mα+c
Chapter 3 Solutions
Single Variable Calculus: Concepts and Contexts, Enhanced Edition
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