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

A lamp stands at the point (4, 9/5) on the edge of the elliptical pond x²/25 + y²/9 = 1, with…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

A lamp stands at the point $\left(4, \dfrac{9}{5}\right)$ on the edge of the elliptical pond $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$ , with lengths in metres. A rail runs along the tangent to the pond at the lamp and a post along the normal. Find the slope of the tangent, the $x$ -coordinate where the tangent meets the $x$ -axis, and the slope of the normal.

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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← Two microphones are at F₁(-5, 0) and F₂(5, 0) on a plan whose lengths are in hundreds of metres…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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