TangentsTangents

(and the discriminant)(and the discriminant)

What is to be learned?What is to be learned?

How to prove tangencyHow to prove tangency Find conditions for tangencyFind conditions for tangency Find points of contact for tangentsFind points of contact for tangents

TangentsTangents

TangentsTangents

Meets at one point onlyMeets at one point only

One solutionOne solution

Using DiscriminantUsing Discriminant

bb22 – 4ac = 0 – 4ac = 0

Show that y = 6x – 9 is tangent Show that y = 6x – 9 is tangent to curve y = xto curve y = x22 and find point of and find point of

contactcontact

Show that Show that y =y = 6x - 9 is tangent 6x - 9 is tangent to curve to curve y =y = x x22 and find point of and find point of

contactcontact y = yy = y

= x= x22

xx22 – 6x + 9 = 0 – 6x + 9 = 0

c.fc.f ax2 + bx + c = 0a = 1, b = -6, c = 9for tangency(-6)2 – 4(1)(9)= 0 as required

Discriminant Must be in form ax2 + bx + c = 0

b2 – 4ac = 0

6x – 96x – 9 – +0

Show that Show that y =y = 66x - 9 is tangent to x - 9 is tangent to curve curve y =y = x x22 and find and find point of point of

contact?contact? y = yy = y

6x – 9 = x6x – 9 = x22

xx22 – 6x + 9 = 0 – 6x + 9 = 0

c.fc.f ax2 + bx + c = 0a = 1, b = -6, c = 9for tangency(-6)2 – 4(1)(9)= 0 as required

Discriminant

Must be in form

ax2 + bx + c = 0

b2 – 4ac = 0

need x and y

Show that Show that y =y = 6x - 9 is tangent to 6x - 9 is tangent to curve curve y =y = x x22 and find and find point of point of

contact?contact? y = yy = y

6x – 9 = x6x – 9 = x22

xx22 – 6x + 9 = 0 – 6x + 9 = 0

(x – 3)(x – 3) = 0(x – 3)(x – 3) = 0

x = 3x = 3

need x and y

Show that Show that y =y = 6x - 9 is tangent to 6x - 9 is tangent to curve curve y =y = x x22 and find and find point of point of

contact?contact? y = yy = y

6x – 9 = x6x – 9 = x22

xx22 – 6x + 9 = 0 – 6x + 9 = 0

(x – 3)(x – 3) = 0(x – 3)(x – 3) = 0

x = 3x = 3

need x and y

Show that Show that y =y = 6x - 9 is tangent to 6x - 9 is tangent to curve curve y =y = x x22 and find and find point of point of

contact?contact? y = yy = y

6x – 9 = x6x – 9 = x22

xx22 – 6x + 9 = 0 – 6x + 9 = 0

(x – 3)(x – 3) = 0(x – 3)(x – 3) = 0

x = 3x = 3

y = 6(3) – 9 or y = 3y = 6(3) – 9 or y = 322

= 9= 9 = 9 = 9need x and y

The Discriminant and The Discriminant and TangencyTangency

Tangents meet curves at Tangents meet curves at one one pointpoint

OneOne solution solution

For tangency For tangency b2 – 4ac = 0

Show that y = 8x - 17 is tangent to Show that y = 8x - 17 is tangent to curve y = xcurve y = x22 - 1 and find point of - 1 and find point of

contactcontact y = yy = y

8x – 17 = x8x – 17 = x22 - 1 - 1

xx22 – 8x + 16 = 0 – 8x + 16 = 0

c.fc.f ax2 + bx + c = 0a = 1, b = -8, c = 16for tangency(-8)2 – 4(1)(16)= 0 as required

Discriminant

Must be in form

ax2 + bx + c = 0

b2 – 4ac = 0

Point of ContactPoint of Contact

UsingUsing x x22 – 8x + 16 = 0 – 8x + 16 = 0

(x – 4)(x – 4) = 0(x – 4)(x – 4) = 0

x = 4x = 4

Using y = xUsing y = x22 – 1 – 1

= 4= 422 – 1 – 1

= 15= 15

PoC (4 , 15)PoC (4 , 15)

y=x2 - 1

y = 8x - 17

(4,15)

x?

y?

Exam Type StuffExam Type Stuff

a) Prove that the line y = 3x + t meets the a) Prove that the line y = 3x + t meets the parabola y = xparabola y = x22 + 4 where x + 4 where x22 – 3x + (4 – t) – 3x + (4 – t) = 0 b) Find t, when line is a = 0 b) Find t, when line is a tangent and P of Ctangent and P of C

a) Prove that the line a) Prove that the line y =y = 3x + t meets the 3x + t meets the parabola parabola y =y = x x22 + 4 where x + 4 where x22 – 3x + (4 – t) – 3x + (4 – t) = 0 b) Find t, when line is a = 0 b) Find t, when line is a tangent and P of Ctangent and P of C

a)Point of Intersectiona)Point of Intersection

y = yy = y

3x + t = x3x + t = x22 + 4 + 4

a) Prove that the line a) Prove that the line y =y = 3x + t meets the 3x + t meets the parabola parabola y =y = x x22 + 4 where + 4 where xx22 – 3x + (4 – t) – 3x + (4 – t) = 0= 0 b) Find t, when line is a b) Find t, when line is a tangent and P of Ctangent and P of C

a)Point of Intersectiona)Point of Intersection

y = yy = y

3x + t = x3x + t = x22 + 4 + 4

0 = x0 = x22 + 4 – 3x – t + 4 – 3x – t

xx22 – 3x + 4 – t = 0 – 3x + 4 – t = 0( ) as requiredQED

a) Prove that the line y = 3x + t meets the a) Prove that the line y = 3x + t meets the parabola y = xparabola y = x22 + 4 where x + 4 where x22 – 3x + (4 – t) – 3x + (4 – t) = 0 = 0 b) Find t, when line is a b) Find t, when line is a tangent and P of Ctangent and P of C

a)Point of Intersectiona)Point of Intersection

y = yy = y

3x + t = x3x + t = x22 + 4 + 4

0 = x0 = x22 + 4 – 3x – t + 4 – 3x – t

xx22 – 3x + 4 – t = 0 – 3x + 4 – t = 0( ) as requiredQED

a) Prove that the line y = 3x + t meets the a) Prove that the line y = 3x + t meets the parabola y = xparabola y = x22 + 4 where x + 4 where x22 – 3x + (4 – t) – 3x + (4 – t) = 0 = 0 b) Find t, when line is a b) Find t, when line is a tangent and P of Ctangent and P of C

b)b) xx22 – 3x + (4 –t) = 0 – 3x + (4 –t) = 0

For tangencyFor tangency

c.f. axc.f. ax22 + bx + c = 0 + bx + c = 0

a = 1, b = -3,a = 1, b = -3,

(-3)(-3)22 – 4(1)(4 – t) = 0 – 4(1)(4 – t) = 0

9 – 4(4 – t) = 09 – 4(4 – t) = 0

9 – 16 + 4t = 09 – 16 + 4t = 0

4t = 7 4t = 7

t = t = 77//44

b2 – 4ac= 0

c = 4 – t

a) Prove that the line y = 3x + t meets the a) Prove that the line y = 3x + t meets the parabola y = xparabola y = x22 + 4 where x + 4 where x22 – 3x + (4 – t) – 3x + (4 – t) = 0 = 0 b) Find t, when line is a b) Find t, when line is a tangent and tangent and P of CP of C

b)b) xx22 – 3x + (4 –t) = 0 – 3x + (4 –t) = 0

For tangencyFor tangency

c.f. axc.f. ax22 + bx + c = 0 + bx + c = 0

a = 1, b = -3,a = 1, b = -3,

(-3)(-3)22 – 4(1)(4 – t) = 0 – 4(1)(4 – t) = 0

9 – 4(4 – t) = 09 – 4(4 – t) = 0

9 – 16 + 4t = 09 – 16 + 4t = 0

4t = 7 4t = 7

t = t = 77//44

b2 – 4ac= 0

c = 4 – t

t = 7/4

a) Prove that the line y = 3x + t meets the a) Prove that the line y = 3x + t meets the parabola y = xparabola y = x22 + 4 where x + 4 where x22 – 3x + (4 – t) – 3x + (4 – t) = 0 = 0 b) Find t, when line is a b) Find t, when line is a tangent and tangent and P of CP of Cb)b) xx22 – 3x + (4 –t) = 0 – 3x + (4 –t) = 0

xx22 – 3x + (4 – – 3x + (4 – 77//44) = 0) = 0xx22 – 3x + – 3x + 99//44 = 0 = 0(x – (x – 33//22)(x – )(x – 33//22)= 0)= 0

x = x = 33//22

y = xy = x2 2 + 4+ 4 = (= (33//22))2 2 + 4+ 4 = = 99//44 + 4 + 4 = 6 ¼ = 6 ¼

t = 7/4

P of C (1½ , 6¼)

a) Prove that the line y = 4x + t meets the a) Prove that the line y = 4x + t meets the parabola y = xparabola y = x22 + 2 where x + 2 where x22 – 4x + (2 – t) – 4x + (2 – t) = 0 b) Find t, when line is a = 0 b) Find t, when line is a tangent and P of Ctangent and P of C

a)Point of Intersectiona)Point of Intersection

y = yy = y

4x + t = x4x + t = x22 + 2 + 2

0 = x0 = x22 + 2 – 4x – t + 2 – 4x – t

xx22 – 4x + 2 – t = 0 – 4x + 2 – t = 0( ) as required

a) Prove that the line y = 4x + t meets the a) Prove that the line y = 4x + t meets the parabola y = xparabola y = x22 + 2 where x + 2 where x22 – 4x + (2 – t) – 4x + (2 – t) = 0 = 0 b) Find t, when line is a b) Find t, when line is a tangent and P of Ctangent and P of C

b)b) xx22 – 4x + (2 –t) = 0 – 4x + (2 –t) = 0

For tangencyFor tangency

c.f. axc.f. ax22 + bx + c = 0 + bx + c = 0

a = 1, b = -4,a = 1, b = -4,

(-4)(-4)22 – 4(1)(2 – t) = 0 – 4(1)(2 – t) = 0

16 – 4(2 – t) = 016 – 4(2 – t) = 0

16 – 8 + 4t = 016 – 8 + 4t = 0

4t = -8 4t = -8

t = -2t = -2

b2 – 4ac= 0

c = 2 – t

a) Prove that the line y = 4x + t meets the a) Prove that the line y = 4x + t meets the parabola y = xparabola y = x22 + 2 where x + 2 where x22 – 4x + (2 – t) – 4x + (2 – t) = 0 = 0 b) Find t, when line is a b) Find t, when line is a tangent and tangent and P of CP of Cb)b) xx22 – 4x + (2 – t) = 0 – 4x + (2 – t) = 0

xx22 – 4x + (2 + 2) = 0 – 4x + (2 + 2) = 0xx22 – 4x + 4 = 0 – 4x + 4 = 0(x – 2)(x – 2)= 0(x – 2)(x – 2)= 0

x = 2x = 2y = xy = x2 2 + 2+ 2 = 2= 22 2 + 2+ 2 = 4 + 2= 4 + 2 = 6 = 6

t = -2

P of C (2 , 6)