Question

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

• A. $2 \sqrt{11}$
• B. $3 \sqrt{2}$
• C. $6 \sqrt{3}$
• D. $8 \sqrt{2}$

Answer 1

The Correct Option is C

$ x^{2} =4y $ $ x-\sqrt{2} y+4\sqrt{2} =0 $ $ x^{2} =4 \left(\frac{x+4\sqrt{2}}{\sqrt{2}}\right) $ $ \sqrt{2}x^{2} +4x+16\sqrt{2} $ $ \sqrt{2}x^{2} - 4x+16\sqrt{2} = 0 $ $x_{1}+ x_{2} =2 \sqrt{2} ; x_{1}x_{2} = \frac{-16\sqrt{2}}{\sqrt{2}} = -16 $ $\left(\sqrt{2}y -4\sqrt{2}\right)^{2} =4y $ $2y^{2} +32 -16y=4y $ $ \ell_{AB} = \sqrt{\left(x_{2} -x_{1}\right)^{2} +\left(y_{2}-y_{1}\right)^{2}} $ $= \sqrt{\left(2\sqrt{2}\right)^{2} +64+\left(10\right)^{2}-4\left(16\right)} $ $ = \sqrt{8+64+100-64} $ $= \sqrt{108} =6\sqrt{3} $