A) an ellipse
B) a hyperbola
C) a parabola
D) a circle
Correct Answer: C
Solution :
Given that \[x={{t}^{2}}+2t-1\] ?(i) and \[y=3t+5\] ?(ii) \[\Rightarrow \] \[t=\frac{y-5}{3}\] On putting the value of t in Eq. (i), we get \[x={{\left( \frac{y-5}{3} \right)}^{2}}+2\left( \frac{y-5}{3} \right)-1\] \[\Rightarrow \]\[x=\frac{1}{9}({{y}^{2}}-10y+25)+\frac{2}{3}(y-5)-1\] \[\Rightarrow \]\[x=\frac{1}{9}({{y}^{2}}-10y+25+6y-30-9)\] \[=\frac{1}{9}({{y}^{2}}-4y-14)\] \[\Rightarrow \]\[9x+14={{y}^{2}}-4y+4-4\] \[\Rightarrow \]\[9x+18={{(y-2)}^{2}}\] \[\Rightarrow \]\[{{(y-2)}^{2}}=9\,(x+2)\] This is an equation of a parabola. Hence, given parametric equations represents a parabola.
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