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Using

$$y=mx\pm a\sqrt{1+m^2}$$

I found $m$ to be $\frac 74$ and $-\frac{4}{7}$. From here I can find the separate equations of the tangent, combin them, simplify the given equation and compare. Clearly, it’s extremely tedious and can’t be done easily. Can I use the fact that the tangents are mutually perpendicular to easy my calculations?

Rodrigo de Azevedo
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Aditya
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    The point $(11,3)$ lies on the tangents. So substituting that into the given equation we get $(11^2+3^2-65)^2+k(11^2+3^2-65)=0$ and hence $k=-(11^2+3^2-65)=-65$. That doesn't seem too tedious. – almagest Jan 26 '20 at 11:41
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    See $$#169$$ of https://archive.org/details/elementsofcoordi00loneuoft/page/142/mode/2up – lab bhattacharjee Jan 26 '20 at 12:07

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