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Question

A tangent having slope of $$-\, \displaystyle \frac{4}{3}$$ to the ellipse $$\displaystyle {\frac{x^2}{18}\, +\, \frac{y^2}{32}\, =\, 1}$$ intersects the major & minor axes in points $$A$$ & $$B$$ respectively. If $$C$$ is the centre of the ellipse then the area of the $$\triangle ABC$$ is


A
12 sq. units
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B
24 sq. units
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C
36 sq. units
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D
48 sq. units
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Solution

The correct option is B $$24$$ sq. units
Since the major axis is along the y-axis.

$$\therefore$$ Equation of tangent is $$x\, =\, my\, +\, \sqrt {b^2m^2\, +\, a^2}$$

Slope of tangent $$=\displaystyle {\frac{1}{m}\, =\, \frac{-4}{3}\, \Rightarrow\, m\, =\, \frac{-3}{4}}$$

Hence, equation of tangent is $$4x + 3y = 24$$ or $$\displaystyle {\frac{x}{6}\, +\, \frac{y}{8}\, =\, 1}$$

Its intercepts on the axes are $$6$$ and $$8$$.

Area $$(\Delta AOB)\, =\, \displaystyle \frac{1}{2}\, \times\, 6\, \times\, 8\, =\, 24$$ sq. unit

Mathematics

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