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# 11X1 T11 06 tangents and normals II

## by Nigel Simmons, Teacher at Baulkham Hills High School on Jul 28, 2010

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## 11X1 T11 06 tangents and normals IIWebinar Transcript

• Tangents & Normals (ii) Using Cartesian
• Tangents & Normals (ii) Using Cartesian (1) Tangent
• Tangents & Normals (ii) Using Cartesian (1) Tangent y x 2  4ay x
• Tangents & Normals (ii) Using Cartesian (1) Tangent y x 2  4ay P( x1 , y1 ) x
• Tangents & Normals (ii) Using Cartesian (1) Tangent y x 2  4ay P( x1 , y1 ) x
• Tangents & Normals (ii) Using Cartesian (1) Tangent x2 y x 2  4ay y 4a P( x1 , y1 ) x
• Tangents & Normals (ii) Using Cartesian (1) Tangent x2 y x 2  4ay y 4a dy x  dx 2a P( x1 , y1 ) x
• Tangents & Normals (ii) Using Cartesian (1) Tangent x2 y x 2  4ay y 4a dy x  dx 2a dy x1 P( x1 , y1 ) when x  x1 ,  dx 2a x
• Tangents & Normals (ii) Using Cartesian (1) Tangent x2 y x 2  4ay y 4a dy x  dx 2a dy x1 P( x1 , y1 ) when x  x1 ,  dx 2a x x1  slope of tangent is 2a
• Tangents & Normals (ii) Using Cartesian (1) Tangent x2 y x 2  4ay y 4a dy x  dx 2a dy x1 P( x1 , y1 ) when x  x1 ,  dx 2a x x1  slope of tangent is 2a x y  y1  1  x  x1  2a
• Tangents & Normals (ii) Using Cartesian (1) Tangent x2 y x 2  4ay y 4a dy x  dx 2a dy x1 P( x1 , y1 ) when x  x1 ,  dx 2a x x1  slope of tangent is 2a x y  y1  1  x  x1  2a 2ay  2ay1  xx1  x12
• Tangents & Normals (ii) Using Cartesian (1) Tangent x2 y x 2  4ay y 4a dy x  dx 2a dy x1 P( x1 , y1 ) when x  x1 ,  dx 2a x x1  slope of tangent is 2a x y  y1  1  x  x1  2a 2ay  2ay1  xx1  x12 2ay  2ay1  xx1  4ay1
• Tangents & Normals (ii) Using Cartesian (1) Tangent x2 y x 2  4ay y 4a dy x  dx 2a dy x1 P( x1 , y1 ) when x  x1 ,  dx 2a x x1  slope of tangent is 2a x y  y1  1  x  x1  2a 2ay  2ay1  xx1  x12 2ay  2ay1  xx1  4ay1 xx1  2a y  y1 
• (2) Normal
• (2) Normal y x 2  4ay x
• (2) Normal y x 2  4ay P( x1 , y1 ) x
• (2) Normal y x 2  4ay P( x1 , y1 ) x
• (2) Normal y x 2  4ay P( x1 , y1 ) x1 1 Show the slope of tangent at P is 2a x
• (2) Normal y x 2  4ay P( x1 , y1 ) x1 1 Show the slope of tangent at P is 2a 2a x 2  slope of normal is  x1
• (2) Normal y x 2  4ay P( x1 , y1 ) x1 1 Show the slope of tangent at P is 2a 2a x 2  slope of normal is  x1  2a y  y1   x  x1  x1
• (2) Normal y x 2  4ay P( x1 , y1 ) x1 1 Show the slope of tangent at P is 2a 2a x 2  slope of normal is  x1  2a y  y1   x  x1  x1 x1 y  x1 y1  2ax  2ax1
• (2) Normal y x 2  4ay P( x1 , y1 ) x1 1 Show the slope of tangent at P is 2a 2a x 2  slope of normal is  x1  2a y  y1   x  x1  x1 x1 y  x1 y1  2ax  2ax1 2ax  x1 y  2ax1  x1 y1
• (3) Line cutting/touching/missing parabola
• (3) Line cutting/touching/missing parabola y x 2  4ay x
• (3) Line cutting/touching/missing parabola y x 2  4ay x
• (3) Line cutting/touching/missing parabola y x 2  4ay x
• (3) Line cutting/touching/missing parabola y x 2  4ay y  mx  b x
• (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x
• (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x
• (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x 2  4amx  4ab  0 x
• (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x 2  4amx  4ab  0 two solutions (cuts) when   0 x
• (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x 2  4amx  4ab  0 two solutions (cuts) when   0 one solution (touches) when   0 x
• (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x 2  4amx  4ab  0 two solutions (cuts) when   0 one solution (touches) when   0 x no solutions (misses) when   0
• (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x 2  4amx  4ab  0 two solutions (cuts) when   0 one solution (touches) when   0 x no solutions (misses) when   0   b2  4ac
• (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x 2  4amx  4ab  0 two solutions (cuts) when   0 one solution (touches) when   0 x no solutions (misses) when   0   b2  4ac   4am   4 1 4ab  2
• (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x 2  4amx  4ab  0 two solutions (cuts) when   0 one solution (touches) when   0 x no solutions (misses) when   0   b2  4ac   4am   4 1 4ab  2  16a 2 m 2  16ab  16a  am 2  b 
• (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x 2  4amx  4ab  0 two solutions (cuts) when   0 one solution (touches) when   0 x no solutions (misses) when   0   b2  4ac   4am   4 1 4ab  2  16a 2 m 2  16ab  16a  am 2  b   two solutions (cuts) when am2  b  0
• (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x 2  4amx  4ab  0 two solutions (cuts) when   0 one solution (touches) when   0 x no solutions (misses) when   0   b2  4ac   4am   4 1 4ab  2  16a 2 m 2  16ab  16a  am 2  b   two solutions (cuts) when am2  b  0 one solution (touches) when am2  b  0
• (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x 2  4amx  4ab  0 two solutions (cuts) when   0 one solution (touches) when   0 x no solutions (misses) when   0   b2  4ac   4am   4 1 4ab  2  16a 2 m 2  16ab  16a  am 2  b   two solutions (cuts) when am2  b  0 one solution (touches) when am2  b  0 (common idea)
• (3) Line cutting/touching/missing parabola y x 2  4ay parabola and tangent meet when; y  mx  b x 2  4a  mx  b  x 2  4amx  4ab  0 two solutions (cuts) when   0 one solution (touches) when   0 x no solutions (misses) when   0   b2  4ac   4am   4 1 4ab  2  16a 2 m 2  16ab  16a  am 2  b   two solutions (cuts) when am2  b  0 one solution (touches) when am2  b  0 (common idea) no solutions (misses) when am2  b  0
• e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2).
• e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b
• e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b
• e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m
• e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m tangents are y  mx  2  3m
• e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m tangents are y  mx  2  3m x2  4 y
• e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m tangents are y  mx  2  3m x2  4 y x 2  4  mx  2  3m 
• e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m tangents are y  mx  2  3m x2  4 y x 2  4  mx  2  3m  x 2  4mx  12m  8  0
• e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m tangents are y  mx  2  3m x2  4 y x 2  4  mx  2  3m  x 2  4mx  12m  8  0 line is a tangent if   0
• e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m tangents are y  mx  2  3m x2  4 y x 2  4  mx  2  3m  x 2  4mx  12m  8  0 line is a tangent if   0  4m   4 112m  8  0 2
• e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m tangents are y  mx  2  3m x2  4 y x 2  4  mx  2  3m  x 2  4mx  12m  8  0 line is a tangent if   0  4m   4 112m  8  0 2 16m2  48m  32  0
• e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m tangents are y  mx  2  3m x2  4 y x 2  4  mx  2  3m  x 2  4mx  12m  8  0 line is a tangent if   0  4m   4 112m  8  0 2 16m2  48m  32  0 m2  3m  2  0  m  1 m  2   0
• e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m tangents are y  mx  2  3m x2  4 y x 2  4  mx  2  3m  x 2  4mx  12m  8  0 line is a tangent if   0  4m   4 112m  8  0 2 16m2  48m  32  0 m2  3m  2  0  m  1 m  2   0 m  1 or m  2
• e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m tangents are y  mx  2  3m x2  4 y x 2  4  mx  2  3m  x 2  4mx  12m  8  0 line is a tangent if   0  4m   4 112m  8  0 2 16m2  48m  32  0 m2  3m  2  0  m  1 m  2   0 m  1 or m  2  tangents are y  x  1 and y  2 x  4
• e.g. Find the equation of the two tangents to the parabola x 2  4 y passing through the point (3,2). tangent will be of the form y = mx + b  2  3m  b b  2  3m tangents are y  mx  2  3m x2  4 y x 2  4  mx  2  3m  x 2  4mx  12m  8  0 Exercise 9G; 1ac, 2ac, 3a, 4, 7, 9, 11, 12, line is a tangent if   0 13, 15, 17, 18  4m   4 112m  8  0 2 16m2  48m  32  0 m2  3m  2  0  m  1 m  2   0 m  1 or m  2  tangents are y  x  1 and y  2 x  4