Geometry-What-are-the-equations-for-two-lines-through-the-origin-that-are-tangent-to-the-ellipse-6-48-5-0-

Question Number 134951 by bobhans last updated on 09/Mar/21

\underset{―}{Geometry}

What are the equations for two lines through the origin that are tangent to the ellipse 6𝑥²−48𝑥+𝑦²+5=0 ?

Answered by EDWIN88 last updated on 09/Mar/21

let the equation tangent line to ellipse

{6 x}^{2} - 48 x + y^{2} + 5 = 0 is y = mx

substitute to ellipse give

{6 x}^{2} - 48 x + m^{2} x^{2} + 5 = 0

(m^{2} + 6) x^{2} - 48 x + 5 = 0, take Δ = 0

\Rightarrow Δ = 48^{2} - 4 (m^{2} + 6) (5) = 0

\Rightarrow 12 \times 48 = 5 (m^{2} + 6)

\Rightarrow m^{2} = \frac{12 \times 48}{5} - 6 = \frac{546}{5}

\Rightarrow m = \pm \sqrt{\frac{546}{5}} . so equation of two line

through the origin that are tangent to the

ellipse {6 x}^{2} - 48 x + y^{2} + 5 = 0 is y = \pm x \sqrt{\frac{546}{5}}

Commented by EDWIN88 last updated on 09/Mar/21

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