Question 1209063
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Find all points having a y-coordinate of -6 whose distance from the point (1,2) is 17. 

(a)By using the Pythagorean Theorem. 

(b)By using the distance formula.
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        In the post by  Edwin,  the solution and the answer are incorrect.

        In his solution,  Edwin mistakenly used  y-coordinate  6  instead of the given  -6.

        His plot is inadequate,  too.


        I came to bring a correct solution.



<pre>
Let's call the given point A = (1,2).


All points having y-coordinate of -6, lie on the horizontal line y = -6.

Vertical distance from the point (1,2) to this line (or simply the distance) is 6+2 = 8 units.

This distance is the length of the perpendicular from this point (1,2) to the line y= -6.


We want to find points C on the line y= -6 such that the distance from A to C is 17 units.


Draw the perpendicular AB from A to line y= -6.  The length of this perpendicular is 8 units.  
The coordinates of B are (1,-6).


The triangle ABC is a right-angled triangle.


Its diagonal AC has the length of 17 units;  its leg AB is of 8 units.


Hence, the leg BC along the line y = -6 is  (Pythagoras)


    {{{sqrt(17^2 -8^2)}}} = {{{sqrt(289-64)}}} = {{{sqrt(225)}}} = 15 units.


Thus possible points C are  (1+15,-6) = (16,-6)  or  (1-15,-6) = (-14,-6).


<U>ANSWER</U>.  There are two such points  C = (16,-6)  and  C' = (-14,-6).
</pre>

Solved &nbsp;(correctly).