Okay so let me just state the question first. Determine the value of the constant real number $k$ so that the tangent lines to the curves $100x-7y^2=0$ and $22x^2 +ky^2=100k+1078$ at the point $(7,10)$ (on both curves) are perpendicual
2026-03-27 10:31:44.1774607504
Tangent lines to curves.
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$m_1 = \dfrac{50}{7y} = \dfrac{50}{7\cdot 10} = \dfrac{5}{7}$, and $m_2 = \dfrac{-44x}{2ky} = \dfrac{-44\cdot 7}{2k\cdot 10} = \dfrac{-77}{5k}$. $l_1 \perp l_2 \rightarrow m_1\cdot m_2 = -1$ gives $k = 11$ as claimed.