For what value of k does the line y = x + k touches the ellipse \(9x^2 + 16y^2\) = 144.

Solution :

\(\because\) Equation of ellipse is \(9x^2 + 16y^2\) = 144 or \(x^2\over 16\) + \({(y-3)}^2\over 9\) = 1

comparing this with \(x^2\over a^2\) + \(y^2\over b^2\) = 1 then we get \(a^2\) = 16 and \(b^2\) = 9

and comparing the line y = x + k with y = mx + c ; m = 1 and c = k

If the line y = x + k touches the ellipse \(9x^2 + 16y^2\) = 144, then \(c^2\) = \(a^2m^2 + b^2\)

\(\implies\) \(k^2\) = 16 \(\times\) \(1^2\) + 9 \(\implies\) \(k^2\) = 25

\(\therefore\) k = \(\pm\)5