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Question: If the normal at any point P on the ellipse cuts the major and minor axes in G and g respectively an...

If the normal at any point P on the ellipse cuts the major and minor axes in G and g respectively and C be the centre of the ellipse, then

A

a2(CG)2+b2(Cg)2=(a2−b2)2a^{2}(CG)^{2} + b^{2}(Cg)^{2} = (a^{2} - b^{2})^{2}

B

a2(CG)2−b2(Cg)2=(a2−b2)2a^{2}(CG)^{2} - b^{2}(Cg)^{2} = (a^{2} - b^{2})^{2}

C

a2(CG)2−b2(Cg)2=(a2+b2)2a^{2}(CG)^{2} - b^{2}(Cg)^{2} = (a^{2} + b^{2})^{2}

D

None of these

Answer

a2(CG)2+b2(Cg)2=(a2−b2)2a^{2}(CG)^{2} + b^{2}(Cg)^{2} = (a^{2} - b^{2})^{2}

Explanation

Solution

Let at point (x1,y1)(x_{1},y_{1}) normal will be (x−x1)x1a2=(y−y1)b2y1\frac{(x - x_{1})}{x_{1}}a^{2} = \frac{(y - y_{1})b^{2}}{y_{1}}

At G, y=0y = 0 ⇒ x=CG=x1(a2−b2)a2x = CG = \frac{x_{1}(a^{2} - b^{2})}{a^{2}} and at g, x=0x = 0⇒

y=Cg=y1(b2−a2)b2y = Cg = \frac{y_{1}(b^{2} - a^{2})}{b^{2}}

x12a2+y12b2=1\frac{x_{1}^{2}}{a^{2}} + \frac{y_{1}^{2}}{b^{2}} = 1 ⇒ a2(CG)2+b(Cg)2=(a2−b2)2a^{2}(CG)^{2} + b(Cg)^{2} = (a^{2} - b^{2})^{2}