First slide

Ellipse

Question

The tangent to the ellipse $3 x^{2} + 16 y^{2} = 12,$ at the point (1, 3/4), intersects the curve $y^{2} + x = 0$ at :

Moderate

A

no point
B

exactly one point
C

two distinct points
D

more than two points

Solution

Tangent at $(1, \frac{3}{4})$ to the ellipse $3 x^{2} + 16 y^{2} = 12$ is
$\begin{array}{l} \frac{x \cdot 1}{4} + \frac{y (\frac{3}{4})}{\frac{3}{4}} = 1 \\ \Rightarrow x + 4 y = 4 \end{array}$
which intersects the curve $y^{2} + x = 0$ at points for which
$y^{2} + (4 - 4 y) = 0 \Rightarrow (y - 2)^{2} = 0 \Rightarrow y = 2$
and the points of intersection is (– 4, 2) exactly one point.

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