Question 554469
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Here is an alternative solution to the first problem which yields nearly the same answer that the other tutor got.  However the solution from the other tutor shows only one of two answers to the question.<br>
The information that the sum of the distances from the two fixed points (-4,0) and (4,0) is 9 is the classical definition of an ellipse with center (0,0) and the two foci at those two points.<br>
The equation of an ellipse with center (0,0), horizontal semi-major axis a and vertical semi-minor axis b is<br>
{{{x^2/a^2+y^2/b^2=1}}}<br>
The distance from the center of the ellipse to each focus is c, where<br> {{{c^2=a^2-b^2}}}<br>
So in this problem we know c is 4.<br>
With the sum of the distances from the two foci being 9, we know that a, the length of the semi-major axis, is 9/2.<br>
Since we know a and c, we can determine b.<br>
{{{c^2=a^2-b^2}}}
{{{4^2=(9/2)^2-b^2}}}
{{{b^2=81/4-16=17/4}}}
{{{b=sqrt(17)/2}}}<br>
Now we know the equation of the ellipse:<br>
{{{x^2/(81/4)+y^2/(17/4)=1}}}<br>
Now substitute the given abscissa x=1 in the equation to find the ordinate y.<br>
{{{1/(81/4)+y^2/(17/4)=1}}}
{{{4/81+4y^2/17=1}}}
{{{4y^2/17=1-4/81=77/81}}}
{{{y^2=(77*17)/(81*4)}}}
{{{abs(y)=sqrt(1309/324)=sqrt(1309)/18}}}<br>
That value to several decimal places is the answer the other tutor got: y = 2.010005835.<br>
But this is an ellipse -- there are two values of y when x is 1.<br>
ANSWERS: 2.010005835 and -2.010005835<br>
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NOTE: For the second problem, see the easy and clear solution from tutor @ikleyn.<br>