Why not 2015?

Geometry Level 5

The sides of a rhombus are parallel to y = 2 x + 3 y=2x+3 and 2 y = x + 5 2y=x+5 . The diagonals of the rhombus intersect at ( 2014 , 2016 ) (2014,2016) . If one vertex of the rhombus lies on the y-axis and the possible values of the ordinates of this vertex are a & b a \ \& \ b such that a > b a>b , then find the value of a b \dfrac{a} {b}


The answer is 2015.

Shubhendra Singh by Shubhendra Singh , 20, India

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1 solution

A diagonal of a rhombus bisects the angle. Slope of the two sides are m 1 = 2 , m 2 = . 5. The slope of the diagonal, m = T a n ( a r c t a n ( m 1 ) + a r c t a n ( m 2 ) 2 ) = 1. When the lower left vertex is on the y-axis, its coordinates are (0,2) \text{A diagonal of a rhombus bisects the angle. Slope of the two sides are }m_1=2,\ m_2=.5.\\ \text{The slope of the diagonal, }m=Tan \left (\dfrac{arctan(m_1)+arctan(m_2)}2 \right )=1.\\ \text{When the lower left vertex is on the y-axis, its coordinates are (0,2)} \\ ∵ it is on the slope 1 and pass through (2014,2016). The upper and lower left verities are at equal vertical distances from (2014,2016). \because \text{ it is on the slope 1 and pass through (2014,2016). }\\ \text{The upper and lower left verities are at equal vertical distances from (2014,2016).}\\ This distance =2016-2=2014. ∴ b = 2 a n d a = 2 + 2 ∗ 2014. a b = 2015 \text{This distance =2016-2=2014.}\\ \therefore \ b=2 \ and \ a=2+2*2014.\\ \dfrac a b = \Large\ \ \ \ \ \ \ \color{#D61F06}{2015}

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I failed to understand how a point lies on the Y- axis with coordinates (2,0). Better check your wording before you post a problem.

Zobair Caca - 5 years, 7 months ago

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Thank you so much. I have corrected. I will try to be more careful.

Niranjan Khanderia - 5 years, 7 months ago

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