Find equation of a line & a perpendicular line, find the intercepts & points of intersection of the 2 lines?

Question

In the equation ax+by+1=0 determine the constants 'a' and 'b' so that the line contains the point (1,0) and has to be perpendicular to the line: x+y=1, need to find the intercepts for the two lines and their points of intersection.

Answer 1

ax + by + 1 passes through (1,0)
putting, x =1, y= 0

a+1 = 0 .. ... ... ... ... (1)

the line is normal to x+y = 1
eqn of a line normal to x+y = 1 is y-x = k (k=a const)
so, y-x-k = 0 & ax + by + 1=0 are identical
=> a/(-1) = b/1 = 1/(-k)
=> a = 1/k , b = -1/k

putting value of a in (1)
1/k = -1
=> k = -1
=> a = -1 , b = 1
so the reqd line is y - x + 1 = 0 (ans)
this has intercept -1 uts in y axis & 1 uts in x axis
x+y=1 has intercepts 1 uts in both axes.
solving for x & y,
adding the eqns,
x+y + y-x+1 = 1
=>y = 0
=> x = 1
so their intersecting pt is (1,0) (ans)

Answer 2

Ok first the easier part. As far as I can tell from the equation, if (1,0) must be in the answer, then the constant "a" must equal -1. So:

-x + by + 1 = 0, or

-x + by = -1

Point of intersection = (1,0), since line 1 has that point (given) and line 2 has that point (1+0=1)

Slope of line 2 (x+y=1) is -1; therefore, slope of a perpendicular line 1 must be +1.

Intercept of line 1 = -1
Intercept of line 2 = +1

Meaning equation of line 1: y = mx + b
y = x - 1
recall that -x + by = -1
by = x - 1

b = 1

So there's the answer. I jumped around a lot to get there so I hope you followed it, but it was easiest (for me) to finish the last part before finishing the first part.

Answer 3

Slope intercept kind fairly makes the working plenty longer. With a sprint prepare you may in many circumstances do right here on your head. Any line perpendicular to 2x - 3y = 12 could have equation 3x + 2y = ok (Swop the coefficients and alter the sign.) Now basically substitute in x = -2, y = 5 to artwork out ok 3*-2 + 2*5 = -6 + 10 = 4 the mandatory line is 3x + 2y = 4

Answer 4

For the first,
y = -(a/b)x - 1/b
for the 2nd,
y = - x + 1
so -(a/b) must be 1
0 = 1*1-1/b
1/b = 1
y = x - 1
the first has x intercept 1 and y intercept - 1
the 2nd has x intercept 1 and y intercept 1
Intersection is at (1,0)