Show that the area of the triangle formed by the tangent and the normal at tha point(a,a) on tha curve y2(2a-x)=x3 and the line x=2a,is 5a2/4.

Equation of the given curve is y2 (2ax) = x3

y2 (2ax) = x3

Differentiating both sides w.r.t  x, we get

Slope of tangent at

∴ Slope of normal at

Equation of tangent at (a, a) is

(ya) = 2 (xa)

y = 2x – 2a + a = 2xa     ...(1)

Equation of normal at (a, a) is

∴ 2y – 2a = – x + a

Equation of the given line is   

x = 2a        ...(3)

Solving (1) and (3), we have

x = 2a and y = 2 (2a) – a = 4aa = 3a

Solving (2) and (3), we have

Solving (1) and (2), we have

x = a and y = a

be the vertices of the triangle.

Are of ∆ABC

Thus, the area of triangle formed is square units.

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