The vertex and the focus of the parabola $2 y^2+5 x-6 y+1=0$ are respectively

• Correct answer
  A

  $\left(\frac{7}{10}, \frac{3}{2}\right),\left(\frac{3}{40}, \frac{3}{2}\right)$
• B

  $\left(\frac{-7}{10}, \frac{3}{2}\right),\left(\frac{53}{40}, \frac{3}{2}\right)$
• C

  $\left(\frac{7}{10}, \frac{-3}{2}\right),\left(\frac{7}{10}, \frac{7}{8}\right)$
• D

  $\left(\frac{-7}{10}, \frac{-3}{2}\right),\left(\frac{7}{10}, \frac{17}{8}\right)$

Solution

Equation of given parabola is $$ \begin{aligned} & 2 y^2+5 x-6 y+1=0 \\ \Rightarrow & 2\left(y^2-3 y\right)+5 x+1=0 \\ \Rightarrow & 2\left(y^2-3 y+\left(\frac{3}{2}\right)^2-\left(\frac{3}{2}\right)^2\right)+5 x+1=0 \\ \Rightarrow & 2\left(y-\frac{3}{2}\right)^2-\frac{9}{2}+5 x+1=0 \\ \Rightarrow & 2\left(y-\frac{3}{2}\right)^2=-5\left(x-\frac{7}{10}\right) \end{aligned} $$
The coordinate of vertex of parabola (i) is $\left(\frac{7}{10}, \frac{3}{2}\right)$ and coordinate of focus of parabola (i) is $$ \begin{aligned} \left(-\frac{5}{8}+\frac{7}{10}, \frac{3}{2}\right) & =\left(\frac{-50+56}{80}, \frac{3}{2}\right) \\ & =\left(\frac{6}{80}, \frac{3}{2}\right) \text { or }\left(\frac{3}{40}, \frac{3}{2}\right) \end{aligned} $$ Hence, option (a) correct.

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