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Mathematics Dictionary
Dr. K. G. Shih

Interception of Curves

Subjects

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Q01 | - How many intersections of y = a*x^2+b*x+c with y = 1/x ?

Q02 | - Use program WebABC find interection of Q01

Q03 | - How to use WebABC ?

Q04 | - Study curve y = 1/x.

Q05 | - Y = 1/x tangents to curve Y = 4*x^2 + k. Find k

Q06 | - Prove that y = 1/x and y = x^2 has only one intersection

Q07 | - Special equations : Intersection of Line and hyperbola

Q08 | - Special equations : Intersection of Line and hyperbola

Q09 | - Special equations : Intersection of circle and hyperbola

Q10 | - How many intersections of y = a*x^2+b*x+c with y = p*x+q ?

Answers

Q01. How many intersections of y = a*x^2+b*x+c with y = 1/x ?
A1. Answer

Q02. Using WebABC find intersections of functions

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Q03. How to use WebABC ?

Q04. Study the curve y = 1/x

* It has vertical asymptote at x=0
* It has hozizontal asymptote at y=0
* The curves are always decreasing (i.e. y' is negative)
* The cuvers are in 1st and 3rd quadrants
* The curve is concave downward if x is less than zero
* The curve is concave upward if x is greater than zero
*

Program 01 25 is graph of y=1/x

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Q05. Y = 1/x tangents to curve Y = 4*x^2 + k. Find k
A5. Solution

* The slope of y=1/x is y'=-1/x^2
* The slope of y=4*x^2+k is y'=8*x
* Since it tangents each other hence slopes of 2 curves same
* Hence 8*x=-1/x^2 or x^3+1/8 = 0
* Hence (x+1/2)*(x^2-x/2+1/4) = 0
* Hence x=-1/2 and y=-2 when the curves tangent each other
* Hence -2 = 4*(-1/2)^2 + k and k = -3
*

Program 09 07 is the graphic solution

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Q06. Prove that y = 1/x and y = x^2 has only one intersection
A6. Solution 1

Q07. Special equations : Intersection of Line and hyperbola

Equations
- x + y = 3 ......... (1)
- x*y = 2 ........... (2)
Solution
- Square (1) - 2*(2), we have (x+y)^2 - 4*x*y = 9 - 8.
- Hence x^2 - 2*x*y + y^2 = 1.
- Hence x - y = 1 ... (3)
- or x - y = -1 ..... (4)
- (1) + (3), we have 2*x = 4 or x = 2 and y = 1.
- (1) + (4), we have 2*x = 2 or x = 1 and y = 2.
- Intersections : (1,2) and (2,1).

Q08. Special equations : Intersection of Line and hyperbola

Equations
- x - y = 1 ......... (1)
- x*y = 2 ........... (2)
Solution
- Square (1) + 4*(2) we have (x-y)^2 + 4*x*y = 1 + 8 = 9.
- Hence x^2 + 2*x*y + y^2 = 9
- Hence x + y = 3 ... (3)
- or x + y = -3 ..... (4)
- (1) + (3), we have 2*x = 4 or x = 2 and y = 1.
- (1) + (4), we have 2*x = -2 or x = -1 and y = -2.
- Intersections : (2,1) and (-1,-2).

Q09. Special equations : Intersection of circle and hyperbola

Equations
- x^2 + y^2 = 4 ......... (1)
- x*y = 1 ............... (2)
Solution
- Square (1) + 2*(2), we have x^2 + 2*x*y + y^2 = 6.
- Hence (x + y) = +Sqr(6) ..............(3)
- Hence (x + y) = -Sqr(6) ..............(4)
- Square (1) - 2*(2), we have x^2 - 2*x*y + y^2 = 2.
- Hence (x - y) = +Sqr(2) ..............(5)
- Hence (x - y) = -Sqr(2) ..............(6)
- Solve (3), (4), (5), (6)
- We have 4 points of Intersections.

Q10. How many intersections of y = a*x^2+b*x+c with y = p*x+q ?
A10. Answer

* Solve p*x+q = a*x^2+b*x+c
* This is a quadratic : a*x^2+(b-p)*x+c+q = 0
* It has no intersection if (b-p)^2 - 4*a*(c-q) less than 0
* It has one intersection if (b-p)^2 - 4*a*(c-q) = 0
* It has two intersections if (b-p)^2 - 4*a*(c-q) > 0
* Using

find intersections of functions

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