Straight lines

Geometry Level 2

If the mirror image of the point (-8,12) with respect to the line 4x+7y+13=0 is (A,B) then 6B-A=


The answer is 4.

Aakhyat Singh by Aakhyat Singh , 20, India

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1 solution

Sundar R
Aug 28, 2017

The slope of given line is -4/7. Therefore, the slope of the line (say line 2) joining (-8,12) to its mirror image is 7/4 (since it is perpendicular to given line , let us call it 1) ,

Line 2) passes through (-8,12)

Equation of line 2) is :
y-12=7/4*(x+8)

4y-48=7x+56

4y=7x+104

Y=7/4x + 26

Thus the coordinates of mirror image = (x, 7x/4+26)

dIst of (-8,12) from line 1) = [(-8 * 4+12 * 7 + 13)/sqrt(-4^2 + 7^2)] = 65/sqrt(65) = sqrt(65)

Dist of (-8,12) from mirror = 2*sqrt(65)

Using euclidean distance formula, (x+8)^2 + (7x/4 + 26 – 12)^2 = 260

X^2 + 16x + 64 + (7x/4 + 14)^2 = 260

X^2+16x + 64 + 49x^2/16 + 196x/4 + 196 = 260

16x^2+256x + 1024 + 49x^2 + 784x + 3136 = 4160

65x^2 + 1040x = 0

65x(x+16)=0

X = -16 = A

Y = 7X/4 + 26 = -28 + 26 = -2 = B

6B – A = -12 + 16 = 4

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Another approach involving matrix algebra and transformations :

  • Find the point of intersection of the line with x-axis which is (-13/4, 0)

  • Translate in the x direction by 13/4 so that the line passes through origin

So (-8,12) goes to (-19/4,12)

The slope of given line is -4/7. So, we rotate by -arctan(-4/7) so that the given line becomes the x-axis, i.e by 0.519146114

If the new coordinates are x', y' , x' = x * cos(t) - y * sin(t) = -19/4 * cos(0.519146114) - 12 * sin(0.519146114) = -10.07782219

y' = x* sin(t) + y* cos(t) = - 19/4 * sin(0.519146114) + 12 * cos(0.519146114) = 8.062257748

Reflect about the x-axis to get ( -10.07782219, - 8.062257748)

Perform the inverse of the previous rotation , i.e rotate by - 0.519146114

x'' = x' * cos(t) - y' * sin(t) = -10.07782219 * cos(-0.519146114) - (- 8.062257748) * (sin(-0.519146114) = -12.75

y''= x' * sin(t) + y" * cos(t) = -10.07782219 * sin(-0.519146114) - 8.062257748 * cos(-0.519146114) = -2

Finally, we translate by -13/4 in the x direction to get the coordinates of the mirror image as (-12.75-3.25,-2) , i.e (-16,-2)

A = -16

B = -2

6A - B = -12 + 16 = 4

One could consolidate all the steps into a single matrix

Sundar R - 3 years, 9 months ago

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