Locus of mid points of chords to the circle `x^2+y^2 -8x +6y+20 =0` which are parallel to the line `3x +4y + 5 =0` is

Options

A. (1,-2)
B. (-1,2)
C. (3,4)
D. (3,-4)

Answer

To find the locus of midpoints of chords to the circle given by the equation \(x^2 + y^2 - 8x + 6y + 20 = 0\) that are parallel to the line \(3x + 4y + 5 = 0\), we can follow these steps:

### Step 1: Rewrite the Circle's Equation
First, we will rewrite the equation of the circle in standard form. The given equation is:

\[ x^2 + y^2 - 8x + 6y + 20 = 0 \]

We can complete the square for \(x\) and \(y\).

For \(x\):
\[ x^2 - 8x = (x - 4)^2 - 16 \]

For \(y\):
\[ y^2 + 6y = (y + 3)^2 - 9 \]

Substituting these back into the equation gives:

\[ (x - 4)^2 - 16 + (y + 3)^2 - 9 + 20 = 0 \]
\[ (x - 4)^2 + (y + 3)^2 - 5 = 0 \]
\[ (x - 4)^2 + (y + 3)^2 = 5 \]

This represents a circle with center \((4, -3)\) and radius \(\sqrt{5}\).

### Step 2: Identify the Slope of the Chords
The line \(3x + 4y + 5 = 0\) can be rewritten in slope-intercept form:

\[ 4y = -3x - 5 \quad \Rightarrow \quad y = -\frac{3}{4}x - \frac{5}{4} \]

The slope of this line is \(-\frac{3}{4}\). Since we are looking for chords that are parallel to this line, the slope of the chords will also be \(-\frac{3}{4}\).

### Step 3: Use the Midpoint Formula
Let the midpoint of the chord be \(P(h, k)\). The equation of the line through the midpoint \(P(h, k)\) with slope \(-\frac{3}{4}\) can be expressed as:

\[ y - k = -\frac{3}{4}(x - h) \]

Rearranging gives:

\[ 3x + 4y - (3h + 4k) = 0 \]

### Step 4: Substitute into the Circle's Equation
The chord will intersect the circle at two points. The condition for the line to be a chord of the circle is that the distance from the center of the circle to the line must be less than or equal to the radius.

Using the formula for the distance from a point \((x_0, y_0)\) to the line \(Ax + By + C = 0\):

\[ \text{Distance} = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]

Substituting \(A = 3\), \(B = 4\), \(C = -(3h + 4k)\), and the center of the circle \((4, -3)\):

\[ \text{Distance} = \frac{|3(4) + 4(-3) - (3h + 4k)|}{\sqrt{3^2 + 4^2}} = \frac{|12 - 12 - 3h - 4k|}{5} = \frac{| -3h - 4k|}{5} \]

### Step 5: Set Up the Condition for the Chord
The distance must be equal to the radius of the circle, which is \(\sqrt{5}\):

\[ \frac{| -3h - 4k|}{5} = \sqrt{5} \]

Multiplying both sides by 5 gives:

\[ | -3h - 4k| = 5\sqrt{5} \]

This can be expressed as two equations:

1. \(-3h - 4k = 5\sqrt{5}\)
2. \(-3h - 4k = -5\sqrt{5}\)

### Step 6: Final Locus Equation
We can simplify one of these equations to find the locus of midpoints. Let's take the first equation:

\[ 3h + 4k = -5\sqrt{5} \]

This represents a line in the \(hk\)-plane.

### Final Answer
The locus of midpoints of chords to the circle that are parallel to the line \(3x + 4y + 5 = 0\) is given by the equation:

\[ 3h + 4k + 5\sqrt{5} = 0 \]