Given the line -3x + 2y = 5, then its slope is ........ Choose the correct answer: (1) The reflection through the y axis of the point (-2, 3) is the point ....... (2) The distance... Given the line -3x + 2y = 5, then its slope is ........ Choose the correct answer: (1) The reflection through the y axis of the point (-2, 3) is the point ....... (2) The distance between P1 = (-2, 3) and P2 = (6, -3) is ........ (3) The midpoint M of the line segment joining P1 = (4, 6) and P2 = (-6, 8) is ....... (4) The equation of a circle with radius 3 and center at (-1, 2) is ....... Read more

Steps to Solve

1. Finding the slope of the line To find the slope of the line given by the equation $-3x + 2y = 5$, we can rearrange the equation into slope-intercept form ($y = mx + b$), where $m$ is the slope.

Starting with the given equation: $$ 2y = 3x + 5 $$

Dividing both sides by 2: $$ y = \frac{3}{2}x + \frac{5}{2} $$

Thus, the slope $m$ is $\frac{3}{2}$.

2. Reflecting a point through the y-axis The reflection of a point $(x, y)$ through the y-axis is given by $(-x, y)$. Thus, the reflection of the point $(-2, 3)$ is $(2, 3)$.

3. Calculating the distance between two points To find the distance between the points $P_1=(-2, 3)$ and $P_2=(6, -3)$, we use the distance formula: $$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

Substituting the coordinates: $$ d = \sqrt{(6 - (-2))^2 + (-3 - 3)^2} $$ $$ = \sqrt{(6 + 2)^2 + (-6)^2} $$ $$ = \sqrt{8^2 + 6^2} $$ $$ = \sqrt{64 + 36} $$ $$ = \sqrt{100} $$ $$ = 10 $$

4. Finding the midpoint of a line segment The midpoint $M$ of a segment joining two points $P_1=(4, 6)$ and $P_2=(-6, 8)$ is calculated using: $$ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$

Substituting the coordinates: $$ M = \left( \frac{4 + (-6)}{2}, \frac{6 + 8}{2} \right) $$ $$ = \left( \frac{-2}{2}, \frac{14}{2} \right) $$ $$ = (-1, 7) $$

5. Writing the equation of a circle The standard form of a circle's equation is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius. Here, the center is $(-1, 2)$ and the radius is $3$. Therefore, the equation is:

$$(x + 1)^2 + (y - 2)^2 = 9$$