Question

if the equation ax² + 5xy - 6y² + 12x + 5y +c = 0 represents a pair of perpendicular straight lines, find a and c​

Answer 1

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Let’s say we have a line

px+qy+r=0px+qy+r=0

then the line perpendicular to it would be

qx−py+s=0qx−py+s=0

To get both at once, multiply

(px+qy+r)(qx−py+s)=0(px+qy+r)(qx−py+s)=0

since this equation is satisfied if and only if at least one of the line equations is satisfied. We can expand the product

pqx2+(q2−p2)xy−pqy2+(ps+qr)x+(qs−pr)y+rs=0pqx2+(q2−p2)xy−pqy2+(ps+qr)x+(qs−pr)y+rs=0

Comparing the y2y2 coefficients, it’s clear

pq=6pq=6

and so comparing the x2x2 coefficients we have

a=6a=6

We also have that

p=6/qp=6/q

and comparing the xyxy coefficients gives us

q2−36/q2=5q2−36/q2=5

q4−5q2−36=0q4−5q2−36=0

q2=9q2=9

q=±3q=±3

and so