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  4. The length of the chord of the parabola x2=4y having equation x-√2y+4√2=0 is

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Q. The length of the chord of the parabola $x^{2}=4y$ having equation $x-\sqrt{2}y+4\sqrt{2}=0$ is

NTA AbhyasNTA Abhyas 2020Conic Sections

Solution:

Solving the equations of the given parabola and the line, we get,
$\left(\sqrt{2} y - 4 \sqrt{2}\right)^{2}=4y\Rightarrow y^{2}-10y+16=0\Rightarrow \left|y_{1} - y_{2}\right|=\sqrt{\left(y_{1} + y_{2}\right)^{2} - 4 y_{1} y_{2}}=\sqrt{100 - 64}=6$
Solution
$\Rightarrow $ Length of the chord $=AB=\sqrt{\left(x_{1} - x_{2}\right)^{2} + \left(y_{1} - y_{2}\right)^{2}}=\sqrt{2 \left(y_{1} - y_{2}\right)^{2} + \left(y_{1} - y_{2}\right)^{2}}=\sqrt{3}\left|y_{1} - y_{2}\right|$
$=6\sqrt{3}$