Using the point-slope form y - y1 = m(x - x1) with m = 2 and (x1, y1) = (1, -3): y - (-3) = 2(x - 1), so y + 3 = 2x - 2, giving y = 2x - 5 or 2x - y - 5 = 0.

y - y1 = m(x - x1)

Slope m = (6 - 2)/(3 - 1) = 4/2 = 2. Using point-slope form with point (1, 2): y - 2 = 2(x - 1), so y - 2 = 2x - 2, giving y = 2x, or 2x - y = 0.

y - y1 = m(x - x1)

Slope of first line m1 = -1/sqrt3; slope of second line m2 = -sqrt3. The angle between them satisfies tan(theta) = |(m1 - m2)/(1 + m1 m2)| = |(-1/sqrt3 + sqrt3)/(1 + (-1/sqrt3)(-sqrt3))| = |((-1 + 3)/sqrt3)/(1 + 1)| = |(2/sqrt3)/2| = 1/sqrt3. So theta = 30 degrees.

tan theta = |(m1 - m2)/(1 + m1 m2)|

The distance of point (x1, y1) from line Ax + By + C = 0 is d = |A x1 + B y1 + C|/sqrt(A^2 + B^2). Here A = 3, B = -4, C = -26 and (x1, y1) = (3, -5). d = |3(3) - 4(-5) - 26|/sqrt(9 + 16) = |9 + 20 - 26|/sqrt25 = |3|/5 = 3/5.

d = |A x1 + B y1 + C|/sqrt(A^2 + B^2)

For parallel lines Ax + By + C1 = 0 and Ax + By + C2 = 0, the distance is d = |C1 - C2|/sqrt(A^2 + B^2). Here A = 3, B = -4, C1 = 7, C2 = 5. d = |7 - 5|/sqrt(9 + 16) = 2/5.

d = |C1 - C2|/sqrt(A^2 + B^2)

The intercept form of a line is x/a + y/b = 1, where a and b are the x- and y-intercepts. Here a = -3 and b = 2, so x/(-3) + y/2 = 1. Multiplying throughout by 6: -2x + 3y = 6, i.e. 2x - 3y + 6 = 0.

x/a + y/b = 1

Midpoint of the segment = ((1 + 3)/2, (1 + 5)/2) = (2, 3). Slope of the segment = (5 - 1)/(3 - 1) = 2, so the slope of the perpendicular bisector = -1/2. Using point-slope form through (2, 3): y - 3 = -1/2 (x - 2), so 2(y - 3) = -(x - 2), 2y - 6 = -x + 2, giving x + 2y - 8 = 0.

product of perpendicular slopes = -1

For the image (h, k) of point (x1, y1) in line Ax + By + C = 0, we use (h - x1)/A = (k - y1)/B = -2(A x1 + B y1 + C)/(A^2 + B^2). Here A = 1, B = 3, C = -7, (x1, y1) = (3, 8). A x1 + B y1 + C = 3 + 24 - 7 = 20, and A^2 + B^2 = 1 + 9 = 10, so the common ratio = -2(20)/10 = -4. Then h = 3 + 1(-4) = -1 and k = 8 + 3(-4) = 8 - 12 = -4. Hence the image is (-1, -4).

(h - x1)/A = (k - y1)/B = -2(A x1 + B y1 + C)/(A^2 + B^2)

Compare with x^2 + y^2 + 2gx + 2fy + c = 0: 2g = -4 so g = -2, 2f = 6 so f = 3, c = -12. Centre = (-g, -f) = (2, -3). Radius = sqrt(g^2 + f^2 - c) = sqrt(4 + 9 + 12) = sqrt25 = 5.

centre (-g, -f), radius sqrt(g^2 + f^2 - c)

The equation of a circle with centre (h, k) and radius r is (x - h)^2 + (y - k)^2 = r^2. Here (h, k) = (2, -3) and r = 5, so (x - 2)^2 + (y + 3)^2 = 25. Expanding: x^2 + y^2 - 4x + 6y - 12 = 0.

(x - h)^2 + (y - k)^2 = r^2

Compare y^2 = 12x with y^2 = 4ax: 4a = 12, so a = 3. The parabola opens to the right. Focus = (a, 0) = (3, 0). Directrix: x = -a, i.e. x = -3. Axis: the x-axis, i.e. y = 0. Length of latus rectum = 4a = 12.

y^2 = 4ax: focus (a,0), directrix x = -a

Here a^2 = 25 (a = 5) and b^2 = 9 (b = 3), with a > b so the major axis is along the x-axis. c = sqrt(a^2 - b^2) = sqrt(25 - 9) = 4. Vertices = (+/-5, 0); foci = (+/-4, 0); eccentricity e = c/a = 4/5; length of latus rectum = 2b^2/a = 2(9)/5 = 18/5.

e = c/a; latus rectum = 2b^2/a

Here a^2 = 16 (a = 4) and b^2 = 9 (b = 3). For a hyperbola, c = sqrt(a^2 + b^2) = sqrt(16 + 9) = sqrt25 = 5. Vertices = (+/-4, 0); foci = (+/-5, 0); eccentricity e = c/a = 5/4. (Length of latus rectum = 2b^2/a = 9/2.)

hyperbola: c = sqrt(a^2 + b^2), e = c/a

The focus is on the positive y-axis and the directrix is y = -3, so the parabola opens upwards with vertex at the origin and form x^2 = 4ay. Here a = 3 (distance from vertex to focus), so x^2 = 4(3)y = 12y.

x^2 = 4ay

The vertices and foci are on the x-axis, so the ellipse is x^2/a^2 + y^2/b^2 = 1 with a > b. From vertices, a = 5; from foci, c = 4. Then b^2 = a^2 - c^2 = 25 - 16 = 9. Hence the equation is x^2/25 + y^2/9 = 1.

b^2 = a^2 - c^2

Latus rectum = 2b^2/a and major axis = 2a. Given 2b^2/a = (1/2)(2a) = a, so 2b^2 = a^2. Since b^2 = a^2(1 - e^2), we get 2a^2(1 - e^2) = a^2, so 2(1 - e^2) = 1, giving 1 - e^2 = 1/2, e^2 = 1/2 and e = 1/sqrt2.

b^2 = a^2(1 - e^2)

The distance between (x1, y1, z1) and (x2, y2, z2) is sqrt[(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2]. Here = sqrt[(4 - 2)^2 + (3 - 3)^2 + (1 - 5)^2] = sqrt[4 + 0 + 16] = sqrt20 = 2 sqrt5.

d = sqrt[(x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2]

The distance of a point (x, y, z) from the origin is sqrt(x^2 + y^2 + z^2). Here = sqrt(1^2 + (-2)^2 + 3^2) = sqrt(1 + 4 + 9) = sqrt14 units.

d = sqrt(x^2 + y^2 + z^2)

By the section formula for internal division in ratio m : n, the coordinates are ((m x2 + n x1)/(m + n), (m y2 + n y1)/(m + n), (m z2 + n z1)/(m + n)) with m = 2, n = 3. x = (2(3) + 3(1))/5 = 9/5; y = (2(4) + 3(-2))/5 = 2/5; z = (2(-5) + 3(3))/5 = -1/5. The point is (9/5, 2/5, -1/5).

((m x2 + n x1)/(m+n), ...)

The centroid of a triangle with vertices (x1,y1,z1), (x2,y2,z2), (x3,y3,z3) is ((x1+x2+x3)/3, (y1+y2+y3)/3, (z1+z2+z3)/3). x = (1+3+4)/3 = 8/3; y = (2-1+0)/3 = 1/3; z = (3+5-2)/3 = 6/3 = 2. Centroid = (8/3, 1/3, 2).

G = ((x1+x2+x3)/3, (y1+y2+y3)/3, (z1+z2+z3)/3)