Question

The length of the chord of contact from point $(2,1)$ to the circle $x^2+y^2+4x+2y+1=0$ is

• A. $\frac{8}{\sqrt{5}}$
• B. $\frac{4}{\sqrt{5}}$
• C. $\frac{4\sqrt{6}}{\sqrt{5}}$
• D. $\frac{2\sqrt{6}}{\sqrt{5}}$

Hint:

Chord of Contact.
For external point, use $T=0$ to find line of chord. Then compute perpendicular from center to line, and apply $2\sqrtR^2 - d^2$.

Answer 1

The Correct Option is A

Chord of contact from $(x_1,y_1)$ to circle $S$: \[ T = 0: xx_1 + yy_1 + g(x + x_1) + h(y + y_1) + c = 0 \] From $S$: $g = 2$, $h = 1$, $c = 1$, and $(x_1, y_1) = (2,1)$: \[ T: 2x + y + 4 + 2 + 1 = 0 \Rightarrow 2x + y + 3 = 0 \] Distance from center $(-2, -1)$ to chord: \[ d = \frac{|2(-2) + (-1) + 3|}{\sqrt{5}} = \frac{2}{\sqrt{5}} \] \[ R = \sqrt{g^2 + h^2 - c} = \sqrt{4 + 1 - 1} = 2 \] Chord length = $2\sqrt{R^2 - d^2} = 2\sqrt{4 - \frac{4}{5}} = \frac{8}{\sqrt{5}}$

Answer 2

Step 1: Identify the circle and the external point
Given circle: \(x^2 + y^2 + 4x + 2y + 1 = 0\)
Point: \(P(2, 1)\)

Step 2: Find the center and radius of the circle
Rewrite the circle in standard form by completing the square:
\[ x^2 + 4x + y^2 + 2y + 1 = 0 \]
Complete the square:
\[ (x^2 + 4x + 4) + (y^2 + 2y + 1) + 1 - 4 - 1 = 0 \implies (x + 2)^2 + (y + 1)^2 = 4 \]
So, center \(C = (-2, -1)\), radius \(r = 2\)

Step 3: Formula for length of chord of contact
Length of chord of contact from point \(P(x_1, y_1)\) to the circle \((x - h)^2 + (y - k)^2 = r^2\) is:
\[ \text{Length} = 2 \sqrt{PT^2 - r^2} \] where \(PT\) is the distance from point \(P\) to center \(C\)

Step 4: Calculate distance \(PT\)
\[ PT = \sqrt{(2 - (-2))^2 + (1 - (-1))^2} = \sqrt{(4)^2 + (2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \]

Step 5: Calculate length of chord of contact
\[ \text{Length} = 2 \sqrt{(2\sqrt{5})^2 - 2^2} = 2 \sqrt{20 - 4} = 2 \sqrt{16} = 2 \times 4 = 8 \]

Step 6: Verify the formula
Alternatively, the length of chord of contact can also be found by:
\[ \text{Length} = \frac{2 \sqrt{S_1}}{\sqrt{1^2 + 1^2}} \] where \(S_1\) is the value of the circle equation at point \(P\).
Calculate \(S_1\):
\[ S_1 = (2)^2 + (1)^2 + 4(2) + 2(1) + 1 = 4 + 1 + 8 + 2 + 1 = 16 \]
Denominator:
\[ \sqrt{4 + 1} = \sqrt{5} \]
Length = \(\frac{2 \sqrt{16}}{\sqrt{5}} = \frac{2 \times 4}{\sqrt{5}} = \frac{8}{\sqrt{5}}\)

Final answer:
\[ \boxed{\frac{8}{\sqrt{5}}} \]