JEE Main · 2021 · 26 Aug 2021 · Shift 1 · Circle

If a line along a chord of the circle

4 x^{2} + 4 y^{2} + 120 x + 675 = 0

, passes through the point

(- 30, 0)

and is tangent to the parabola

y^{2} = 30 x

, then the length of this chord is:

(A)7
(B)5
(C) $53$
(D) $35$ ✓ Correct

Answer: (D)

Solution

Dividing by

4

, the circle is

(x + 15)^{2} + y^{2} = \frac{225}{4}

, with centre

(- 15, 0)

and radius

\frac{15}{2}

.
For

y^{2} = 30 x

the parameter is

a = \frac{15}{2}

, so a tangent of slope

m

y = m x + \frac{15}{2 m}

. Passing through

(- 30, 0)

requires

30 m = \frac{15}{2 m}

, so

m^{2} = \frac{1}{4}

; taking

m = \frac{1}{2}

, the chord lies along

x - 2 y + 30 = 0

.
Its distance from the centre is

p = \frac{∣ - 15 - 0 + 30 ∣}{1 + 4} = \frac{15}{5} = 35,

so the half-chord is

\frac{225}{4} - 45 = \frac{3 5}{2}

and the chord length is

35

JEE Main · 2021 · 26 Aug 2021 · Shift 1 · Single correct option

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