Show that the area of the triangle formed by the tangent and the normal at tha point(a,a) on tha curve - Maths - Application of Derivatives

Question

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

Lalit Mehra · Accepted Answer

Equation of the given curve is y2 (2a â€“ x) = x3 y2 (2a â€“ x) = x3 Differentiating both sides w r t x, we get $y^{2} \times (- 1) + (2 a - x) \times 2 y \frac{d y}{d x} = 3 x^{2} \Rightarrow (2 a - x) 2 y \frac{d y}{d x} = 3 x^{2} + y^{2} \Rightarrow \frac{d y}{d x} = \frac{3 x^{2} + y^{2}}{2 y (2 a - x)}$ Slope of tangent at $(a, a), m_{1} = \frac{3 a^{2} + a^{2}}{2 a (2 a - a)} = \frac{4 a^{2}}{2 a^{2}} = 2$ âˆ´ Slope of normal at $(a, a), m_{2} = \frac{- 1}{2}$ Equation of tangent at (a, a) is (y â€“ a) = 2 (x â€“ a) âˆ´ y = 2x â€“ 2a + a = 2x â€“ a (1) Equation of normal at (a, a) is $(y - a) = \frac{- 1}{2} (x - a)$ âˆ´ 2y â€“ 2a = â€“ x + a $\Rightarrow y = \frac{- x}{2} + \frac{3 a}{2} (2)$ Equation of the given line is x = 2a (3) Solving (1) and (3), we have x = 2a and y = 2 (2a) â€“ a = 4a â€“ a = 3a Solving (2) and (3), we have $x = 2 a and y = \frac{- 2 a}{2} + \frac{3 a}{2} = \frac{a}{2}$ Solving (1) and (2), we have x = a and y = a $Let A (2 a, 3 a), B (2 a, \frac{a}{2}) and C (a, a)$ be the vertices of the triangle Are of âˆ†ABC $= \frac{1}{2} | 2 a (\frac{a}{2} - a) + 2 a (a - 3 a) + a (3 a - \frac{a}{2}) | = \frac{1}{2} | 2 a \times (\frac{- a}{2}) + 2 a \times (- 2 a) + a \times \frac{5 a}{2} | = \frac{1}{2} | - a^{2} - 4 a^{2} + \frac{5 a^{2}}{2} | = \frac{1}{2} | - {5 a}^{2} + \frac{5 a^{2}}{2} | = \frac{1}{2} | \frac{- 5 a^{2}}{2} | = \frac{5 a^{2}}{4}$ Thus, the area of triangle formed is $\frac{5 a^{2}}{4}$ square units

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.

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 5a^2/4.