# 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}$$=2 at the point (3,2) \begin{array}{1 1}\text{True}\\\text{False}\end{array} ## 1 Answer Comment A) Toolbox: • Condition for a line y=mx+c to be a tangent to an ellipse is c=\sqrt{b^2+a^2m^2} Answer : True Equation of the given ellipse is \large\frac{x^2}{9}+\frac{y^2}{4}$$=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$
$c=\sqrt{a^2m^2+b^2}$
$4=\sqrt{18\times (-\large\frac{2}{3})^2\normalsize +8}$
$\Rightarrow \sqrt{16}=4$