The value of k such that the line y=2x+k touc | vedclass

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(A) The condition for the line $y=mx+c$ to be a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is $c^2=a^2m^2+b^2$.Given the ellipse equat

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$\pm \sqrt{23}$

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(A) The condition for the line $y=mx+c$ to be a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is $c^2=a^2m^2+b^2$.Given the ellipse equation $3x^2+5y^2=15$,we divide by $15$ to get the standard form: $\frac{x^2}{5}+\frac{y^2}{3}=1$.Here,$a^2=5$,$b^2=3$,$m=2$,and $c=k$.Substituting these values into the condition $c^2=a^2m^2+b^2$:$k^2 = (5)(2^2) + 3$$k^2 = 5 	imes 4 + 3$$k^2 = 20 + 3 = 23$$k = \pm \sqrt{23}$.

The value of $k$ such that the line $y=2x+k$ touches the ellipse $3x^2+5y^2=15$ is

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