Question

If the straight line through the point p(3,4) makes an angle π/6 with the x - axis and meets the line 3x+5y+1=0 at Q the length of PQ is a(3√3-b) then a/b is​

Answer 1

Answer:

explanation

Step-by-step explanation:

Approximately 5.11777 units long.

Following all the radicals in this problem gets very complicated so I have chosen to use decimal approximations. The tangent of 30 degrees is 1/(radical 3), I will approximate this as 0.57735.

The first line has a slope of 0.57735 because tangent is the change in y coordinates divided by the change in x coordinates, which is also the tangent. Start with point slope form ( Y - y1 = m(X - x1)) and change it to slope intercept form ( y = mx + b)

y - 4 = 0.57735(x - 3) distribute the 0.57735.

y - 4 = 0.57735 x - 1.73205 add 4 on both sides to isolate y.

y = 0.57735 x + 2.26795

Now take the second equation isolate y and then we set the 2 y values equal to each other.

12 x + 5 y + 10 = 0 subtract 12 x and 10 from both sides.

5 y = -12 x - 10 divide both sides by 5, y will now be isolated.

y = -2.4 x - 2 Now set both expressions for y equal to each other.

0.57735 x + 2.26795 = -2. 4 x - 2 Add 2.4 x to both sides.

2.97735 x + 2.26795 = - 2 Subtract 2.26795 to both sides.

2.97735 x = -4.26795 Divide both sides by 2.97735

x = -1.43347271…. Approximate this as x = -1.43347

Now find y by putting in -1.43347 as the value of x.

y = -2.4(-1.43347) - 2 Simplifies to y = 1.440328.

So point Q is ( -1.43347, 1.440328)

Now apply the distance formula d = square root of ( (x2 - x1)^2 + (y2 - y1)^2)

square root of ((3 - -1.43347)^2 + (4 - 1.44347)^2)

square root of ( 19.65565624 + 6.535845641)

5.11776336…. approximate as 5.11777

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