HardMathematicsHigh School (11–12)Coordinate Geometry › Circles and Conic SectionsNumerical

Lines of slope 1 are drawn so that they just touch the ellipse x²/16 + y²/9 = 1, and lines of slope…

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Lines of slope $1$ are drawn so that they just touch the ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$ , and lines of slope $2$ so that they just touch the hyperbola $\dfrac{x^2}{9} - \dfrac{y^2}{16} = 1$ . Find the positive $y$ -intercept of the tangent to the ellipse. Find also the positive $y$ -intercept of the tangent to the hyperbola.

Need a reminder? The key ideaShow

The tangent at $(x_1, y_1)$ is $\dfrac{xx_1}{a^2} \pm \dfrac{yy_1}{b^2} = 1$ . With slope $m$ : $c^2 = a^2m^2 \pm b^2$ . The director circle is $x^2 + y^2 = a^2 \pm b^2$ . For $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ the sign of $h^2 - ab$ names the conic when $\Delta \ne 0$ .

The worked solution

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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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