Show that the curves x 2 a 2- 1- y 2 b 2- 1-1 and x 2 a 2- 2- y 2 b 2- 2-1 intersect at right angles-

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Byju's Answer

Standard XII

Mathematics

Angle of Intersection of Two Curves

Show that the...

Question

Show that the curves $\frac{x^{2}}{a^{2} + λ_{1}} + \frac{y^{2}}{b^{2} + λ_{1}} = 1 and \frac{x^{2}}{a^{2} + λ_{2}} + \frac{y^{2}}{b^{2} + λ_{2}} = 1$ intersect at right angles.

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Solution

$We have, \frac{x^{2}}{a^{2} + λ_{1}} + \frac{y^{2}}{b^{2} + λ_{1}} = 1 . . . (1) and \frac{x^{2}}{a^{2} + λ_{2}} + \frac{y^{2}}{b^{2} + λ_{2}} = 1 . . . (2) Now we can find the slope of both the curve by differentiating w . r . t x \Rightarrow \frac{2 x}{a^{2} + λ_{1}} + \frac{2 y \frac{d y}{d x}}{b^{2} + λ_{1}} = 0 and \frac{2 x}{a^{2} + λ_{2}} + \frac{2 y \frac{d y}{d x}}{b^{2} + λ_{2}} = 0 \Rightarrow \frac{d y}{d x} = - \frac{x}{y} \times \frac{b^{2} + λ_{1}}{a^{2} + λ_{1}} and \frac{d y}{d x} = - \frac{x}{y} \times \frac{b^{2} + λ_{2}}{a^{2} + λ_{2}} \Rightarrow m_{1} = - \frac{x}{y} \times \frac{b^{2} + λ_{1}}{a^{2} + λ_{1}} and m_{2} = - \frac{x}{y} \times \frac{b^{2} + λ_{2}}{a^{2} + λ_{2}} Subtracting (2) from (1), we get, x^{2} (\frac{1}{a^{2} + λ_{1}} - \frac{1}{a^{2} + λ_{2}}) + y^{2} (\frac{1}{b^{2} + λ_{1}} - \frac{1}{b^{2} + λ_{2}}) = 0 \Rightarrow \frac{x^{2}}{y^{2}} = \frac{λ_{2} - λ_{1}}{(b^{2} + λ_{1}) (b^{2} + λ_{2})} \times \frac{1}{\frac{λ_{1} - λ_{2}}{(a^{2} + λ_{1}) (a^{2} + λ_{2})}} N o w, m_{1} \times \times m_{2} = \frac{x^{2}}{y^{2}} \times \frac{b^{2} + λ_{1}}{a^{2} + λ_{1}} \times \frac{b^{2} + λ_{2}}{a^{2} + λ_{2}} = \frac{λ_{2} - λ_{1}}{(b^{2} + λ_{1}) (b^{2} + λ_{2})} \times \frac{(a^{2} + λ_{1}) (a^{2} + λ_{2})}{λ_{1} - λ_{2}} \times \frac{b^{2} + λ_{1}}{a^{2} + λ_{1}} \times \frac{b^{2} + λ_{2}}{a^{2} + λ_{2}} = - 1 hence, (1) and (2) cuts orthogonally .$

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Show that the curves x2a2+λ1+y2b2+λ1=1 and x2a2+λ2+y2b2+λ2=1intersect at right angles.

Show that the curves $\frac{x^{2}}{a^{2} + λ_{1}} + \frac{y^{2}}{b^{2} + λ_{1}} = 1 and \frac{x^{2}}{a^{2} + λ_{2}} + \frac{y^{2}}{b^{2} + λ_{2}} = 1$ intersect at right angles.