Find the values of r for which the line (r + 5)x – (r – 1)y + r 2 + 4r − 5 = 0 is [2] (a) parallel to the x-axis. (b) parallel to the line 2x − y + 4 = 0. (c) parallel to the y-axis. (d) perpendicular to the line x − y + 32 = 0

Q: A laser beam shines along the ray given by r(t) = (5.1.7) + (5,3,-2) for t≥ 0. The second laser beam…

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Q: The line in the direction of i −3j + 4k through the point (1, 0, −2).

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Q: Find the intersection point of 2x – y = 5 and 4x = – 2y + 6

A: Given: 2x - y = 5, 4x = -2y + 6 Calculations: From first equation, 2x - 5 = y Substitute this in…

Q: Solve for fx(-9, 9) and fy(-9, 9). f(x, y) = 5x2 + 8xy + 4y2 + 6x − 9y; (−9, 9)

A: Given:fx,y=5x2+8xy+4y2+6x-9yTo find:→fx-9,9→fy-9,9

Q: Find the x- and y-intercepts of x+7y=−7.

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Q: y=-x² +6x-2.

A: Given function is: f(x)=-x2+6x-2 which is a quadratic function, whose graph is a parabola.

Q: If F is the point (- 3, 5) and G is the point (0. - 4), then an equation of FG is: a. y = - x + 4 b.…

A: F is the point (-3,5) and G is the point (0,-4)

Q: y-intercepts of 7x2 + 7y2 + 9x − 8y = 1

A: Given equation 7x2 + 7y2 + 9x − 8y = 1

Q: qx-5=(3p-2)x-2r find x in terms of p q and r

A: Solved below

Q: Find an equation of the line that passes through the point  (3, −1)  and is parallel to the line…

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Q: find the y intercept of y=5log(x+3) -5

A: Given: y=5log(x+3) -5

Q: Solve for y in terms of x   7x+3y=2

A: We first subtract 7x from both sides    7x+3y=2 7x-7x+3y=2-7x 3y=2-7x

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.