Question

Write the length of the chord of the parabola y² = 4ax which passes through the vertex and is inclined to the axis at \[\frac{\pi}{4}\] 

Answer 1

Let OP be the chord.
Let the coordinates of P be  \[\left( x_1 , y_1 \right)\] 

From the figure, we have: 

\[O P^2 = {x_1}^2 + {y_1}^2\] 

And, \[\tan\frac{\pi}{4} = \frac{y_1}{x_1}\] 

\[\Rightarrow x_1 = y_1\]                         (2) 

Also,  \[\left( x_1 , y_1 \right)\] lies on the parabola. 

∴ \[{y_1}^2 = 4a x_1                              (3)

Using (2) and (3), we get: 

\[{x_1}^2 = 4a x_1 \Rightarrow x_1 = 4a\]           ...(4) 

∴ From (4), (1) and (2), we have: 

\[O P^2 = \left( 4a \right)^2 + \left( 4a \right)^2 = 32 a^2 \]
\[ \Rightarrow OP = 4\sqrt{2}a\] 

Therefore, the length of the chord is \[4\sqrt{2}a \text{ units }\]