Paranormal Parabola?

I assume by seeing the word 'Parabola', the problem will be attempted by only those who know something about Parabolas, hence also must be acquainted to Conics.

If you don't know about Conics or have forgotten terms like Axis, Directrix, Focus and Vertex,etc. You can refer to this wiki .

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• Every conic (on Earth ;)) can be defined if you have its Focus and its Directrix. So the next natural question must be do we have it? We have the Focus but we need the Directrix. Now, Directrix is a line and to define a line uniquely, it must satisfy two given conditions. Like it must pass through two given points. Or its slope must be given and a point through which it passes , etc.

• The Vertex is the mid point of the line joining Focus and the Foot Of The Perpendicular Of The Axis(of the Parabola) On The Directrix(of the Parabola). Therefore using section formula we can find the Foot Of The Perpendicular Of The Axis On The Directrix. Lets denote Foot Of The Perpendicular Of The Axis On The Directrix by Q. Then, also notice that Q lies on the Directrix. So, after we determine Q, we will be left to find just one more condition to find the equation of The Directrix. If we denote by $\displaystyle \left( h,k \right)$ the Foot Of The Perpendicular(Q), then $\displaystyle \cfrac { h+1 }{ 2 } =2\Rightarrow h=3$ and $\displaystyle \cfrac { k-1 }{ 2 } =1\Rightarrow k=3$ . $\displaystyle \therefore \left( h,k \right) \equiv \left( 3,3 \right)$ . Now if we had one more condition, say slope of the Directrix was known, then its equations could be written as: $\displaystyle y-k=m\left( x-h \right)$ , where $\displaystyle m$ is its slope. So, can we find its slope? Or can we find any other condition to determine it uniquely? Lets see.

• Now, slope of the axis is defined since Focus and Vertex is given. And the axis passes through both of them. So slope of the axis is $\displaystyle \cfrac { 1-(-1) }{ 2-1 } =2$ . Then slope of the Directrix must be $\displaystyle -\cfrac { 1 }{ 2 }$ , since Directrix is $\displaystyle \bot$ to the Axis. Now, we do have the value of $\displaystyle m=-\cfrac { 1 }{ 2 }$ .Therefore equation of the Directrix must be $\displaystyle y-3=-\cfrac { 1 }{ 2 } \left( x-3 \right) \Rightarrow x+2y-9=0$ .

• By the definition of a Parabola, if any point $\displaystyle \left( x,y \right)$ lies on our Parabola then it must satisfy this equation :- $\displaystyle \sqrt { { \left( x-1 \right) }^{ 2 }+{ \left( y+1 \right) }^{ 2 } } =\cfrac { \left| x+2y-9 \right| }{ \sqrt { { 1 }^{ 2 }+{ 2 }^{ 2 } } }$ . Which on further simplification and modification can be written as $\displaystyle 4{ x }^{ 2 }-4xy+{ y }^{ 2 }+8x+46y-71=0$ . And on comparing it to the equation:- $\displaystyle a{ x }^{ 2 }+2hxy+{ by }^{ 2 }+2gx+2fy+c=0$ we get the values of: $\displaystyle a=4;\quad \quad 2h=-4\Rightarrow h=-2;\quad \\ b=1;\quad \quad 2g=8\Rightarrow g=4\\ c=-71;\quad \quad 2f=46\Rightarrow f=23$ . Hence the value of $\displaystyle Z=a+h+b+g+f+c=-41$ .