If the straight line through the point p(3,4)
makes an angle with the x-axis in the
positive direction and meets the line
3x+5y+1=0 at Q and if PQ =
k (3V3 – 5) then k =
Answers
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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