Question

Find the equations of the chords of the parabola y2 = 4ax which pass through the point (–6a, 0) and which subtends an angle of 45° at the vertex.

Answer 1

Answer on Question #56981 – Math – Analytic Geometry

Question

Find the equations of the chords of the parabola $y^{2} = 4ax$ which pass through the point $(-6a, 0)$ and which subtends an angle of $45{}^{\circ}$ at the vertex.

Solution

The vertex of parabola

$$y^{2} = 4ax$$

is $(0,0)$.

Equation of a chord in the slope-intercept form is

$$y = kx + b$$

Here $k = \tan 45{}^{\circ} = 1$, because a chord subtends an angle of $45{}^{\circ}$ at the vertex.

So, in fact (1) is given by

$$y = x + b$$

On the other hand, this line passes through the point $(-6a, 0)$, consequently its coordinates satisfy equation (2):

$$0 = -6a + b,$$ $$b = 6a.$$

Finally, $y = x + 6a$ is the equation of chord.

Answer: $y = x + 6a$

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