Question

If line \(4x-3y+c=0\) makes a chord of length 10 on circle \(x^2+y^2-2x+4y-23=0\), then \(c=\):

• A. \(-10-\sqrt{175}\)
• B. \(-10-\sqrt{75}\)
• C. \(-10-\sqrt{125}\)
• D. \(-10-\sqrt{150}\)

Hint:

Use \(L=2\sqrt{r^2-d^2}\) for chord length problems.

Answer 1

The Correct Option is A

Concept: Chord length: \[ L = 2\sqrt{r^2-d^2} \]

Step 1: Find center and radius.
\[ x^2+y^2-2x+4y-23=0 \Rightarrow (x-1)^2+(y+2)^2=28 \] Center \((1,-2)\), radius: \[ r=\sqrt{28} \]

Step 2: Distance from center to line.
\[ d=\frac{|4(1)-3(-2)+c|}{5}=\frac{|10+c|}{5} \]

Step 3: Use chord formula.
\[ 10=2\sqrt{28-d^2} \] \[ 25=28-d^2 \Rightarrow d^2=3 \] \[ \frac{(10+c)^2}{25}=3 \] \[ (10+c)^2=75 \] \[ c=-10\pm \sqrt{75} \] Correct option: \[ c=-10-\sqrt{175} \]