If $A(2,-3)$ and $C(-6,7)$ are opposite vertices of a rhombus ABCD, then the equation of diagonal BD is

Question

If $ x = a(\cos \theta + \theta \sin \theta), \, y = a(\sin \theta - \theta \cos \theta) $, then $ \frac{d^2 y}{dx^2} $ equals

Answer 1

Step 1: Use the locus idea
Every point on BD is equally far from A and C, so BD is the perpendicular bisector of AC.

Step 2: Set distances equal
$(x-2)^2+(y+3)^2=(x+6)^2+(y-7)^2$. Expanding: $-4x+6y+13=12x-14y+85$, so $16x-20y+72=0$, i.e. $4x-5y+18=0$. Option (C).

Final Answer:
BD is $4x-5y+18=0$, option (C). \[ \boxed{4x-5y+18=0} \]

If \(A(2,-3)\) and \(C(-6,7)\) are opposite vertices of a rhombus ABCD, then the equation of diagonal BD is

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The Correct Option is C

Solution and Explanation

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