Radius of the largest circle which passes through the focus of the...

Radius of the largest circle which passes through the focus of the parabola

y^{2} = 4x

and contained in it, is Radius of the largest circle which passes through the focus of the parabola y^ 2 = 4x and contained in it, is

Options

A $8$
B $4$
C $2$
D $5$

Correct answer

B. $4$

Step-by-step solution

Let $r$ be the radius of the largest circle passing through the focus \left(1, 0\right) of y^{2} = 4x $<img src="/api/proxy-image?url=https%3A%2F%2Ffirebasestorage.googleapis.com%2Fv0%2Fb%2Fquantrexacademy-app.firebasestorage.app%2Fo%2Fquestions%252Ffigs%252Fpyq%252Fbitsat%252F807de1a8-d722-452e-8c32-2df7ef662bcd-d48.png%3Falt%3Dmedia&clean=1&v=wm2" data-qx-orig-src="https://firebasestorage.googleapis.com/v0/b/quantrexacademy-app.firebasestorage.app/o/questions%2Ffigs%2Fpyq%2Fbitsat%2F807de1a8-d722-452e-8c32-2df7ef662bcd-d48.png?alt=media" data-qx-storage-src="https://firebasestorage.googleapis.com/v0/b/quantrexacademy-app.firebasestorage.app/o/questions%2Ffigs%2Fpyq%2Fbitsat%2F807de1a8-d722-452e-8c32-2df7ef662bcd-d48.png?alt=media" style="width: 325px; height: 250px;" />Clearly, centre of the circle will be on$ X $-axis and its coordinates are \left(1 + r, 0\right)$ .

The equation of the circle, is

\left(x - 1 - r\right)2 + y^{2} = r^{2} $.It touches$ y^{2} = 4x $. Therefore, the equation \left(x - r - 1\right)2 + 4x = r^{2}$ must have equal roots.

$∴ 4\left(1 - r\right)2 - 4\left(2r + 1\right) = 0$

$⇒ r = 4$

Hence, option (b) is correct.

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