Let L be a tangent line to the parabola y^{2} | vedclass

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(B) The equation of the parabola is $y^{2} = 4(x-5)$.The equation of the tangent to the parabola $y^{2} = 4a(x-h)$ at $(x_{1}, y_{1})$ is $yy_{1} = 2a

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$14$

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(B) The equation of the parabola is $y^{2} = 4(x-5)$.The equation of the tangent to the parabola $y^{2} = 4a(x-h)$ at $(x_{1}, y_{1})$ is $yy_{1} = 2a(x+x_{1}) - 4ah$.Here,$a=1$,$h=5$,$x_{1}=6$,and $y_{1}=2$. Substituting these values:$2y = 2(x+6) - 20$$2y = 2x + 12 - 20$$2y = 2x - 8$$y = x - 4$,which can be written as $x - y - 4 = 0$.The condition for the line $y = mx + c$ to be a tangent to the ellipse $\frac{x^{2}}{A^{2}} + \frac{y^{2}}{B^{2}} = 1$ is $c^{2} = A^{2}m^{2} + B^{2}$.Here,$m = 1$,$c = -4$,$A^{2} = 2$,and $B^{2} = b$.Substituting these into the condition:$(-4)^{2} = 2(1)^{2} + b$$16 = 2 + b$$b = 14$.

List-$I$	List-$II$
$(A)$ $\left[\frac{p}{2}\left(t+\frac{1}{t}\right), \frac{q}{2}\left(t-\frac{1}{t}\right)\right]$	$(I)$ parabola
$(B)$ $(p+q \cos \theta, r+q \sin \theta)$	$(II)$ circle
$(C)$ $(p+\lambda^2, q-\lambda)$	$(III)$ ellipse
	$(IV)$ hyperbola

Let $L$ be a tangent line to the parabola $y^{2}=4x-20$ at the point $(6,2)$. If $L$ is also a tangent to the ellipse $\frac{x^{2}}{2}+\frac{y^{2}}{b}=1$,then the value of $b$ is equal to ..... .

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