The shortest distance between the line
Question:

The shortest distance between the line $x-y=1$ and the curve $x^{2}=2 y$ is:

  1. (1) $\frac{1}{2}$

     

  2. (2) 0

  3. (3) $\frac{1}{2 \sqrt{2}}$

  4. (4) $\frac{1}{\sqrt{2}}$


Correct Option:

Solution:

Shortest distance must be along common normal

$m_{1}($ slope of line $x-y=1)=1 \Rightarrow$ slope of perpendicular line $=-1 m_{2}=\frac{2 x}{2}=x \Rightarrow m_{2}=h \Rightarrow$

slope of normal $-\frac{1}{h}$

$-\frac{1}{h}=-1 \Rightarrow h=1$

sopoint is $\left(1, \frac{1}{2}\right)$

$\mathrm{D}=\left|\frac{1-\frac{1}{2}-1}{\sqrt{1+1}}\right|=\frac{1}{2 \sqrt{2}}$

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