What is the perpendicular bisector equation of the chord y equals x plus 5 within the circle x2 plus 4x -18y plus 59 equals 0?

Answer 1

x² + 4x - 18y + 59 = 0 is not a circle; it can be rearranged into:

y = (x² + 4x + 59)/18

which is a parabola. You have missed out a y² term.

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Assuming you meant: x² + 4x + y² - 18y + 59 = 0, then:

The perpendicular bisector of a chord passes through the centre of the circle.

The slope m' of a line perpendicular to another line with slope m is given by m' = -1/m

The chord y = x + 5 has slope m = 1 → the perpendicular bisector has slope m' = -1/1 = -1

A circle with centre Xc, Yc and radius r has an equation in the form:

(x - Xc)² + (y - Yc)² = r²

The equation given for the circle can be rearrange into this form by completing the square in x and y:

x² + 4x + y² - 18y + 59 = 0

→ (x + (4/2))² - (4/2)² + (y - (18/2))² - (18/2)² + 59 = 0

→ (x + 2)² +(y - 9)² - 2² - 9² + 59 = 0

→ (x + 2)² + (y - 9)² = 4 + 81 - 59

→ the circle has centre (-2, 9)

(The radius, if wanted, is given by r² = 4 + 81 - 59 = 36 = 6²)

The equation of a line with slope m' through a point (Xc, Yc) has equation:

y - Yc = m'(x - Xc)

→ y - 9 = -1(x - -2)

→ y - 9 = -x - 2

→ y + x = 7

The perpendicular bisector of the chord y = x + 5 within the circle x² + 4x + y² - 18y + 59 = 0 is y + x = 7

Answer 2

The given equation for a circle is incorrect and so therefore it follows that finding the perpendicular bisector equation of the chord is not possible

Answer 3

The equation of the perpendicular bisector of the chord is x + y = 19. However, the equation which is supposed to be of a circle is actually that of a parabola.