# SOLUTION: Find the length of the chord of the ellipse {(x^2)/(a^2)}+{(y^2)/(b^2)}=1 directed along the diagonal of the rectangle constructed on the axes of the ellipse?

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 Question 558308: Find the length of the chord of the ellipse {(x^2)/(a^2)}+{(y^2)/(b^2)}=1 directed along the diagonal of the rectangle constructed on the axes of the ellipse? Answer by Edwin McCravy(9719)   (Show Source): You can put this solution on YOUR website!``` We need to find the distance between the two points where the green line intersects the red ellipse. First find the equation of the green line: It passes through the origin (0,0) and the point (a,b) We find its slope: m = m = = Using the point-slope formula: y - y1 = m(x - x1) y - 0 = (x - 0) y = x To find the endpoints of the chord, where the green line intersects the red ellipse, we solve the system of equations: We clear each of fractions: b²x² + a²y² = a²b² ay = bx Square both sides of the second equation: a²y² = b²x² Substitute b²x² for a²y² in b²x² + b²x² = a²b² 2b²x² = a²b² x² = x² = x = x = x = Substitute a²y² for b²x² in a²y² + a²y² = a²b² 2a²y² = a²b² y² = y² = y = y = y = So the end points of the chord are and We use the distance formula to find the length of the chord: d = d = d = d = d = d = That's it. Edwin```