Answer to Question #224018 in Analytic Geometry for Bless

Question

 Find the equations of the tangents to the ellipse x² + 4y² = 9 which are parallel to the line 2x + 3y = 0. 

Answer 1

$\text{We know that the equation of the tangent with slope m to the ellipse} \\\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 (1) \\\text{If the line $y = mx + c$ touches the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$,} \\\text{then $c^2 = a^2m^2 + b^2$.} \\\text{is $y=mx \pm \sqrt{a^2m^2+b^2} \,(2)$}\\\text{The equation of the ellipse is}\\x^2+4y^2=9,\\1=\frac{x^2}{9}+\frac{y^2}{\frac{9}{4}}.\\ \text{Comparing this with equation(1), we have that}\\a^2=9,b^2=\frac{9}{4} \\\text{From 2x+3y=0, we have that $y=-\frac{2}{3}$} \\\implies m = -\frac{2}{3} \\\text{Using (2), the required equations of tangent are} \\y = -\frac{2}{3}x \pm \sqrt{9.\frac{4}{9}+\frac{9}{4}}\\\implies y=-\frac{2}{3}x \pm \frac{5}{2} \\\text{Multiplying the above equation by 6} \\\implies 6y=-4x \pm 15$