Question 1204346
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The answer from tutor @Mananth is incorrect.  There is an error somewhere in her work.<br>
The method she shows is valid:
(1) Find the equation of the line through the two given points;
(2) Find the equation of the line perpendicular to that given line passing through (1,-4);
(3) Find the point of intersection of the two lines; and
(4) find the distance between (1,-4) and that point of intersection<br>
Do all those steps correctly and you will get the right answer.<br>
But there is a much faster way to solve the problem.<br>
There is an easy formula for finding the shortest distance from a given point to a given line.<br>
If the equation of the given line is Ax+By+C=0 (the equation must be in that form), and the given point is (p,q), then the shortest distance from the point to the line is<br>
{{{(Ap+Bq+C)/sqrt(A^2+B^2)}}}<br>
For the given problem, the equation of the line through (-5,2) and (3,4), in the required form, is<br>
{{{x-4y+13=0}}}<br>
Then, with the given point (1,-4), the shortest distance from the point to the line is<br>
{{{(1(1)-4(-4)+13)/sqrt(1^2+4^2)=30/sqrt(17)}}}<br>
That gives the shortest distance as 7.276, to 3 decimal places.<br>