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Questions  >>  CBSE XI  >>  Math  >>  Conic Sections
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Q)

State whether the following statement is True or False : The line $2x+3y=12$ touches the ellipse $\large\frac{x^2}{9}+\frac{y^2}{4}$$=2$ at the point $(3,2)$

$\begin{array}{1 1}\text{True}\\\text{False}\end{array} $

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A)
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  • Condition for a line $y=mx+c$ to be a tangent to an ellipse is $c=\sqrt{b^2+a^2m^2}$
Answer : True
Equation of the given ellipse is $\large\frac{x^2}{9}+\frac{y^2}{4}$$=2$
$\Rightarrow \large\frac{x^2}{18}+\frac{y^2}{8}$$=1$
Here $a^2=18$ and $b^2=8$
Equation of the given line is $2x+3y=12$
Slope of the line $m=-\large\frac{2}{3}$
$c=\large\frac{12}{3}$$=4$
$c=\sqrt{a^2m^2+b^2}$
$4=\sqrt{18\times (-\large\frac{2}{3})^2\normalsize +8}$
$\Rightarrow \sqrt{16}=4$
LHS=RHS
Hence the statement is True.
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