Two common tangents to the circle {x^2} + {y^ | vedclass

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(B) The equation of any tangent to the parabola ${y^2} = 8ax$ is given by $y = mx + \frac{2a}{m}$.This line is also a tangent to the circle ${x^2} + {

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$y = \pm (x + 2a)$

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(B) The equation of any tangent to the parabola ${y^2} = 8ax$ is given by $y = mx + \frac{2a}{m}$.This line is also a tangent to the circle ${x^2} + {y^2} = 2{a^2}$.The perpendicular distance from the center $(0, 0)$ of the circle to the line $mx - y + \frac{2a}{m} = 0$ must be equal to the radius $r = \sqrt{2}a$.Thus,$\frac{|\frac{2a}{m}|}{\sqrt{m^2 + 1}} = \sqrt{2}a$.Squaring both sides,we get $\frac{4a^2}{m^2(m^2 + 1)} = 2a^2$.This simplifies to $m^2(m^2 + 1) = 2$,or $m^4 + m^2 - 2 = 0$.Factoring the quadratic in $m^2$,we have $(m^2 - 1)(m^2 + 2) = 0$.Since $m^2$ must be real and positive,we have $m^2 = 1$,which gives $m = \pm 1$.Substituting $m = \pm 1$ into the tangent equation $y = mx + \frac{2a}{m}$,we get $y = \pm (x + 2a)$.

Two common tangents to the circle ${x^2} + {y^2} = 2{a^2}$ and the parabola ${y^2} = 8ax$ are

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