Answer to Question #236241 in Analytic Geometry for luka

Question #236241

Given point A (2 , -3) in a plane; find the coordinates of its image:

a) Under enlargement about ( -3 , 1 )with scale factor 2; (3marks)

b) Under rotation of center ( -2 , 1 ) and angle θ=π/6


1
Expert's answer
2021-09-22T13:40:27-0400

a) Distance from centre of enlargement (3,1)( -3 , 1 ) to point A(2,3)A (2 , -3) is is multiplied by scale factor 2


(x1(3))=2(2(3))=>x1=7(x_1-(-3))=2(2-(-3))=>x_1=7

(y11)=2(31)=>y1=7(y_1-1)=2(-3-1)=>y_1=-7

A1(7,7)A_1(7, -7)


b)

(x+2)2+(y1)2=32(x+2)^2+(y-1)^2=32

y=x1y=-x-1


Points A(2,3)A(2,-3) and A2(x2,y2)A_2(x_2, y_2) lie on the circle

(x+2)2+(y1)2=32(x+2)^2+(y-1)^2=32

Point A(2,3)A(2,-3) is the point of intersection this circle and the line

y=x1y=-x-1

Rotation of center (2,1)( -2 , 1 ) and angle θ=π/6.θ=π/6.

The image of the line y=x1y=-x-1 is the line


y=tan(π4π6)x+by=-\tan(\dfrac{\pi}{4} -\dfrac{\pi}{6})x+b

y=tan(π12)x+(12tan(π12))y=-\tan(\dfrac{\pi}{12})x+(1-2\tan(\dfrac{\pi}{12}))

Point A2(x2,y2)A_2(x_2,y_2) is the point of intersection the circle

(x+2)2+(y1)2=32(x+2)^2+(y-1)^2=32

and the line

y=tan(π12)x+(12tan(π12))y=-\tan(\dfrac{\pi}{12})x+(1-2\tan(\dfrac{\pi}{12}))


Then


y2=tan(π12)x2+(12tan(π12))y_2=-\tan(\dfrac{\pi}{12})x_2+(1-2\tan(\dfrac{\pi}{12}))

(x2+2)2+(y21)2=32(x_2+2)^2+(y_2-1)^2=32

Substitute

(x2+2)2+(tan(π12)x22tan(π12))2=32,(x_2+2)^2+(-\tan(\dfrac{\pi}{12})x_2-2\tan(\dfrac{\pi}{12}))^2=32,

x2>0x_2>0

x22+4x2+4+tan2(π12)x22+4tan(π12)x2x_2^2+4x_2+4+\tan^2(\dfrac{\pi}{12})x_2^2+4\tan(\dfrac{\pi}{12})x_2

+4tan2(π12)=32+4\tan^2(\dfrac{\pi}{12})=32

sec2(π12)x22+4(1+tan(π12))x2\sec^2(\dfrac{\pi}{12})x_2^2+4(1+\tan(\dfrac{\pi}{12}))x_2

+4sec2(π12)32=0+4\sec^2(\dfrac{\pi}{12})-32=0

D=16+32tan(π12)+16tan2(π12)16sec4(π12)D=16+32\tan(\dfrac{\pi}{12})+16\tan^2(\dfrac{\pi}{12})-16\sec^4(\dfrac{\pi}{12})


+128sec2(π12)+128\sec^2(\dfrac{\pi}{12})

Since x2>0x_2>0

x2=2cos2(π12)+cos2(π12)D4x_2=-2\cos^2(\dfrac{\pi}{12})+\cos^2(\dfrac{\pi}{12})\sqrt{\dfrac{D}{4}}

y2=tan(π12)x2+(12tan(π12))y_2=-\tan(\dfrac{\pi}{12})x_2+(1-2\tan(\dfrac{\pi}{12}))


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