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Ellipse

Section: Mathematics

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Question : 65 of 117

Marks: +1, -0

If two tangents drawn from a point

(\alpha, \beta)

lying on the ellipse

25 x^2+4 y^2=1

to the parabola

y^2=4 x

are such that the slope of one tangent is four times the other, then the value of

(10\alpha+5)^2+(16\beta^2+50)^2

[24-Jun-2022-Shift-1]

Your Answer:

Solution: 👈: Video Solution

\because (\alpha, \beta)

lies on the given ellipse,

25\alpha^2+4\beta^2=1 \;\; . . .(i)

Tangent to the parabola,

y=m x+\;\frac{1}{m}

passes through

(\alpha, \beta)

. So,

\alpha m^2-\beta m+1=0

has roots

m_1

and

4 m_1

m_1+4 m_1=\;\frac{\beta}{\alpha}

and

m_1 \cdot 4 m_1=\;\frac{1}{\alpha}

Gives that

4\beta^2=25\alpha \;\; . . .(ii)

from (i) and (ii)

25(\alpha^2+\alpha)=1 \;\; . . .(iii)

Now,

(10\alpha+5)^2+(16\beta^2+50)^2

=25(2\alpha+1)^2+2500(2\alpha+1)^2

=2525(4\alpha^2+4\alpha+1)

from equation (iii)

=2525\left(\;\frac{4}{25}+1\right)

=2929

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