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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