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If two tangents drawn from a point (a, b) lying on the ellipse 25x² + 4y² = 1 to the parabola y² = 4x are such that the slope of one tangent is four times the other, then the value of ${(10 α + 5)}^{2} + {(16 β^{2} + 50)}^{2}$ equals __________.

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alok kumar singh | Contributor-Level 10

2 months ago

$\frac{x^{2}}{\frac{1}{25}} + \frac{y^{2}}{\frac{1}{4}} = 1$

$25 α^{2} + 4 β^{2} = 1 . . . . . . . . . . (i)$

Equation of tangent to parabola y = mx + $\frac{1}{m}$ passes

though (a, b)

$α m^{2} - β m + 1 = 0$

$m_{1} + m_{2} = \frac{β}{α}, 4 m_{1}^{2} = \frac{1}{α}$

$∴ f r o m (i) & (i i)$

25(a² + a) = 1 …………(iii)

$∴ {(10 α + 5)}^{2} + {(16 β^{2} + 50)}^{2} = 2929$

$∴ a = 1 + 0 + b {(s i n - \frac{π}{2})}^{\frac{π}{2}} + 2 b + 1$

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Slope of axis = $\frac{1}{2}$

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⇒ 2y – x – 4 = 0

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If two tangents drawn from a point (a, b) lying on the ellipse 25x² + 4y² = 1 to the parabola y² = 4x are such that the slope of one tangent is four times the other, then the value of ${(10 α + 5)}^{2} + {(16 β^{2} + 50)}^{2}$ equals __________.

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If two tangents drawn from a point (a, b) lying on the ellipse 25x2 + 4y2 = 1 to the parabola y2 = 4x are such that the slope of one tangent is four times the other, then the value of ( 1 0 α + 5 ) 2 + ( 1 6 β 2 + 5 0 ) 2 equals __________.

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If two tangents drawn from a point (a, b) lying on the ellipse 25x² + 4y² = 1 to the parabola y² = 4x are such that the slope of one tangent is four times the other, then the value of ${(10 α + 5)}^{2} + {(16 β^{2} + 50)}^{2}$ equals __________.