JEE Main · 2023 · 01 Feb 2023 · Shift 2 · Parabola

If the x-intercept of a focal chord of the parabola

y^{2} = 8 x + 4 y + 4

is 3, then the length of this chord is equal to ______.

Answer: 16

Solution

Completing the square,

y^{2} - 4 y = 8 x + 4

gives

(y - 2)^{2} = 8 (x + 1)

, a parabola with vertex

(- 1, 2)

and

4 a = 8

, so

a = 2

and focus

(1, 2)

. A focal chord with

x

-intercept

3

passes through the focus

(1, 2)

and the point

(3, 0)

, so its slope is

\frac{0 - 2}{3 - 1} = - 1

, an inclination of

4 5^{\circ}

to the axis. The length of a focal chord inclined at angle

θ

\frac{4 a}{sin ^{2} θ}

; with

4 a = 8

and

sin^{2} 4 5^{\circ} = \frac{1}{2}

this equals

\frac{8}{1/2} = 16

. The length of the chord is

16

JEE Main · 2023 · 01 Feb 2023 · Shift 2 · Integer answer

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