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Q.
The order of the differential equation of all tangent lines to the parabola $ y={{x}^{2}} $ is
JamiaJamia 2013
Solution:
The parametric form of the given equation is $ x=t $ and $ y={{t}^{2}} $ . The equation of any tangent at t is $ 2xt=y+{{t}^{2}} $ On differentiating, we get $ 2t={{y}_{1}} $ On putting this value in the equation of tangent, we have $ \frac{2x{{y}_{1}}}{2}=y+{{\left( \frac{{{y}_{1}}}{2} \right)}^{2}} $ $ \Rightarrow $ $ 4x{{y}_{1}}=4y+y_{1}^{2} $ So, the order of this equation is one.