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  • Tangents are drawn from the points on the line <img alt="x-y-5=0 \; \text{to}\; x^2+4 y^2=4" src="https://entrancecorner.oncodecogs.com/gif.latex?x-y-5%3D0%20%5C%3B%20%5Ctext%7Bto%7D%5C%3

Tangents are drawn from the points on the line x-y-5=0 \; \text{to}\; x^2+4 y^2=4 then all the chords of contact passes through a fixed point, whose coordinates are

Option: 1

\left(\frac{4}{5},-\frac{1}{5}\right)


Option: 2

\left(\frac{4}{5},\frac{1}{5}\right)


Option: 3

\left(-\frac{4}{5},\frac{1}{5}\right)


Option: 4

None 


Answers (1)

best_answer

(a) Let A\left(x_1, x_1-5\right) be a point on x-y-5=0, then chord of contact of x^2+4 y^2=4 wrt A is

\begin{array}{ll} & x x_1+4 y\left(x_1-5\right)=4 \\ \\\Rightarrow \quad & (x+4 y) x_1-(20 y+4)=0 \end{array}

It is passes through a fixed point.

\\\therefore \quad x+4 y=0 \text { and } 20 y+4=0\\ \\ (\because \text{from} P+\lambda Q=0)

\Rightarrow \quad y=-\frac{1}{5} \text { and } x=\frac{4}{5}

The coordinates of fixed point are \left(\frac{4}{5},-\frac{1}{5}\right)

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Rishi

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