The number of real tangents that can be drawn to the ellipse 3x^2 + 5y^2 =32 and 25x^2 +9y^2 = - Maths - Conic Sections

Question

The number of real tangents that can be drawn to the ellipse
3x^2 + 5y^2 =32 and 25x^2 +9y^2 = 450 passing through the point
(3,5) is ?

Ayush Jha · Accepted Answer

We know that equation of tangent to an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ ,

$y = m x + \sqrt{a^{2} m^{2} + b^{2}}$

So the equation of the given ellipse can be written as-

$25 x^{2} + 9 y^{2} = 450 \frac{x^{2}}{18} + \frac{y^{2}}{50} = 1$

The tangent passes through a point (3,5). So equation of tangent is-

$5 = 3 m + \sqrt{18 m^{2} + 50}$

On squaring we get,

$11 m^{2} + 30 m + 25 = 0$

So, discriminant of the above quadratic equation is-

$D = 900 - 1100 = - 200 \Rightarrow D < 0$

Hence no real value of m exists. So no real tangent can be drawn.

Similarly do the other part as well.

The number of real tangents that can be drawn to the ellipse 3x^2 + 5y^2 =32 and 25x^2 +9y^2 = 450 passing through the point (3,5) is ?