A) ellipse
B) parabola
C) circle
D) hyperbola
Correct Answer: D
Solution :
Equation of normal is\[y-y=-\frac{dx}{dy}(X-x)\] it meets the x-axis at G. Therefore, coordinates of\[G=\left( x+y\frac{dy}{dx},0 \right)\] According to question, \[\left| x+y\frac{dy}{dx} \right|=|2x|\Rightarrow y\frac{dy}{dx}=x\]or\[y\frac{dy}{dx}=-3x\] \[\Rightarrow \] \[ydy=xdx\] Or \[ydy=-3xdx\] \[\Rightarrow \] \[\frac{{{y}^{2}}}{2}=\frac{{{x}^{2}}}{2}+C\] Or \[\frac{{{y}^{2}}}{2}=-\frac{3{{x}^{2}}}{2}+C\] \[\Rightarrow \]\[{{x}^{2}}-{{y}^{2}}=-2C\]or\[3{{x}^{2}}+{{y}^{2}}=2C\]
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