The values that m can take,so that the straig | vedclass

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(C) The line $y=4x+m$ is tangent to the ellipse $x^2+4y^2=4$. Substituting $y=4x+m$ into the equation of the ellipse,we get: $x^2+4(4x+m)^2=4$ $x^2+4(

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$\pm \sqrt{65}$

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(C) The line $y=4x+m$ is tangent to the ellipse $x^2+4y^2=4$. Substituting $y=4x+m$ into the equation of the ellipse,we get: $x^2+4(4x+m)^2=4$ $x^2+4(16x^2+8mx+m^2)=4$ $x^2+64x^2+32mx+4m^2-4=0$ $65x^2+32mx+4(m^2-1)=0$ Since the line is tangent to the curve,the discriminant $D$ must be zero: $D = (32m)^2 - 4(65)(4(m^2-1)) = 0$ $1024m^2 - 16(65)(m^2-1) = 0$ Divide by $16$: $64m^2 - 65(m^2-1) = 0$ $64m^2 - 65m^2 + 65 = 0$ $-m^2 + 65 = 0$ $m^2 = 65$ $m = \pm \sqrt{65}$

The values that $m$ can take,so that the straight line $y=4x+m$ touches the curve $x^2+4y^2=4$ are

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