Q88 / 160 · AP EAMCET 2020 (21 Sep Shift 1) · Mathematics

The parabola with directrix \(x+2 y-1=0\) and focus \((1,0)\) is

  • Correct answer
    A
    \(4 x^2-4 x y+y^2-8 x+4 y+4=0\)
  • B
    \(4 x^2+4 x y+y^2-8 x+4 y+4=0\)
  • C
    \(4 x^2+4 x y+y^2+8 x-4 y+4=0\)
  • D
    \(4 x^2-4 x y+y^2-8 x-4 y+4=0\)

Solution

Equation of parabola having directrix \(x+2 y-1=0\) and focus \((1,0)\) is \(\begin{aligned} & \frac{|x+2 y-1|}{\sqrt{1+4}}=\sqrt{(x-1)^2+(y-0)^2} \\ \Rightarrow \quad & (x+2 y-1)^2=5\left[(x-1)^2+y^2\right] \\ \Rightarrow \quad & x^2+4 y^2+1+4 x y-2 x-4 y=5 \\ & \quad\left(x^2+y^2-2 x+1\right) \\ \Rightarrow \quad & 4 x^2-4 x y+y^2-8 x+4 y+4=0 \end{aligned}\) Hence, option (a) is correct.

Hence, the correct answer is option: 'A'.

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