Question

Find the equation of the tangent to ellipse at the point of the ellipse $5x^2 + 3y^2 = 137$ whose ordinate is 2.

Answer 1

The equations of the tangents are $25x + 6y = 137$ and $-25x + 6y = 137$.

Answer 2

1. Substitute $y=2$ into the ellipse equation $5x^2 + 3y^2 = 137$ to find the x-coordinates: $5x^2 + 3(2)^2 = 137$ $5x^2 + 12 = 137$ $5x^2 = 125$ $x^2 = 25$ $x = \pm 5$ The points on the ellipse are $(5, 2)$ and $(-5, 2)$.

2. Use the general formula for the tangent to an ellipse $Ax^2 + By^2 = C$ at a point $(x_1, y_1)$, which is $Axx_1 + Byy_1 = C$. For the ellipse $5x^2 + 3y^2 = 137$, we have $A=5$, $B=3$, $C=137$.

3. For the point $(5, 2)$: $5x(5) + 3y(2) = 137$ $25x + 6y = 137$

4. For the point $(-5, 2)$: $5x(-5) + 3y(2) = 137$ $-25x + 6y = 137$

Thus, the equations of the tangents are $25x + 6y = 137$ and $-25x + 6y = 137$.