For the ellipse \(\frac{x^2}{18}+\frac{y^2}{32}=1\), if a tangent with slope \(\frac{-4}{3}\) intersects the…
For the ellipse \(\frac{x^2}{18}+\frac{y^2}{32}=1\), if a tangent with slope \(\frac{-4}{3}\) intersects the major and minor axes at \(P\) and \(Q\) respectively. Find \(P\) and \(Q\).
\(P(0,8), Q(6,0)\)
\(P(0,6), Q(8,0)\)
\(P(3 \sqrt{2}, 0), Q(0,4 \sqrt{2})\)
\(P(0,3 \sqrt{2}), Q(4 \sqrt{2}, 0)\)
Solution
Equation of tangent to the ellipse \(\frac{x^2}{18}+\frac{y^2}{32}=1\) having slope \(-\frac{4}{3}\) is
\(\begin{array}{rlrl}
& & & y=-\frac{4}{3} x \pm \sqrt{18\left(\frac{16}{9}\right)+32} \\
\Rightarrow & & y & =-\frac{4}{3} x \pm 8 \\
\Rightarrow & 4 x+3 y & = \pm 24
\end{array}\)
The tangent \(4 x+3 y=24\) cuts the major and minor axes at point \(P(0,8)\) and \(Q(6,0)\).
And the tangent \(4 x+3 y=-24\) cuts the major and minor axes at point \(P(0,-8)\) and \(Q(-6,0)\).