Question

2. The equation of a circle is x2 + y2 – 4x + 6y – 77 = 0.
a) Find the x-coordinates of the points A and B where the circle intersects the x-axis.
b) Find point of intersection of the tangents to the circle at A and B.​

Answer 1

a) The x-coordinates of the points A and B are $x_{A} = 11$ and $x_{B} = -7$.

b) The point of intersection of the tangents to the circle at A and B is $(x,y) = (2, 27)$.

a) In this part we simplify the equation of the circle and solve for $x$ to determine the coordinates associated to the required points:

The coordinates of the two points so that circle intersects the x-axis, occurs when the following condition is met:

$f(x, 0) = 0$ (1)

Where $f(x,y)$ is in implicit form.

If we know that the equation of the circle is $x^{2} + y^{2} - 4\cdot x + 6\cdot y - 77 = 0$ and $y = 0$, then the reduced equation is:

$x^{2} - 4\cdot x - 77 = 0$ (2)

And we obtain the x-coordinates of the points by factoring (2):

$(x-11) \cdot (x + 7) = 0$

The x-coordinates of the points A and B are $x_{A} = 11$ and $x_{B} = -7$.

b) Lines are defined by both slope and intercept. Slope is defined by implicit differentiation and intercept is found by equation of the line.

Given that $A(x,y) = (11, 0)$ and $B(x,y) = (-7, 0)$, we determine line equations corresponding to each tangent line:

Slope

$2\cdot x + 2\cdot y \cdot \dot y - 4 + 6\cdot \dot y = 0$

$(2\cdot y + 6)\cdot \dot y = 4 - 2\cdot x$

$\dot y = \frac{2 - x}{y + 3}$

Point A

$\dot y = \frac{2-11}{0 + 3}$

$\dot y = -3$

Point B

$\dot y = \frac{2-(-7)}{0 + 3}$

$\dot y = 3$

Intercept

Point A ($A(x,y) = (11, 0)$, $\dot y = -3$)

$b = y - m\cdot x$

$b = 0 - (-3)\cdot (11)$

$b = 33$

Point B ($B(x,y) = (-7, 0)$, $\dot y = 3$)

$b = y - m\cdot x$

$b = 0 - 3\cdot (-7)$

$b = 21$

The equations of the lines tangent to the circle are:

$y = -3\cdot x +33$ (3)

$y = 3\cdot x +21$ (4)

The point of intersection is the solution of this system of linear equations:

$(x,y) = (2, 27)$

The point of intersection of the tangents to the circle at A and B is $(x,y) = (2, 27)$.

We kindly invite you to see this question on circles: https://brainly.com/question/23988015