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

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