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Mathematics Question on Parabola

The length of the chord of the parabola x2=4yx^2 = 4y having equation x−2y+42=0x - \sqrt{2} y + 4 \sqrt{2} = 0 is :

A

2112 \sqrt{11}

B

323 \sqrt{2}

C

636 \sqrt{3}

D

828 \sqrt{2}

Answer

636 \sqrt{3}

Explanation

Solution

x2=4yx^{2} =4y x−2y+42=0x-\sqrt{2} y+4\sqrt{2} =0 x2=4(x+422)x^{2} =4 \left(\frac{x+4\sqrt{2}}{\sqrt{2}}\right) 2x2+4x+162\sqrt{2}x^{2} +4x+16\sqrt{2} 2x2−4x+162=0\sqrt{2}x^{2} - 4x+16\sqrt{2} = 0 x1+x2=22;x1x2=−1622=−16x_{1}+ x_{2} =2 \sqrt{2} ; x_{1}x_{2} = \frac{-16\sqrt{2}}{\sqrt{2}} = -16 (2y−42)2=4y\left(\sqrt{2}y -4\sqrt{2}\right)^{2} =4y 2y2+32−16y=4y2y^{2} +32 -16y=4y ℓAB=(x2−x1)2+(y2−y1)2\ell_{AB} = \sqrt{\left(x_{2} -x_{1}\right)^{2} +\left(y_{2}-y_{1}\right)^{2}} =(22)2+64+(10)2−4(16)= \sqrt{\left(2\sqrt{2}\right)^{2} +64+\left(10\right)^{2}-4\left(16\right)} =8+64+100−64= \sqrt{8+64+100-64} =108=63= \sqrt{108} =6\sqrt{3}