Question

Let the lines (y−2)=m₁ (x−5) and (y+4)=m₂ (x−3) intersect at right angles at P (where m1 and m2 are parameters). If locus of P is, x²+y²+gx+fy+7=0, then |f+g|=?

Answer 1

To find the value of |f+g|, we need to determine the relationship between the slopes of the intersecting lines and substitute them into the equation of the locus. By finding the negative reciprocal slope, we can determine the value of |f+g|, which is equal to 1/m₂.

Step-by-step explanation:

In this question, we are given two lines represented by the equations (y−2)=m₁(x−5) and (y+4)=m₂(x−3). The lines intersect at a point P and they are perpendicular to each other. We are also given that the locus of P is represented by the equation x²+y²+gx+fy+7=0.

To find the value of |f+g|, we need to determine the relationship between the parameters m₁ and m₂, and then substitute these values into the equation of the locus.

Since the lines are perpendicular, their slopes must be negative reciprocals of each other.

So, we have:

1. m₁ * m₂ = -1
2. m₁ = -1/m₂

Substituting the value of m₁ into the equation of the locus:

x²+y²+gx+f(-1/m₂)x+7=0

Now we need to compare this equation with the given equation x²+y²+gx+fy+7=0.

Comparing the coefficients of corresponding terms in both equations:

1. f = -1/m₂
2. g = 0

Therefore, |f+g| = |-1/m₂ + 0| = |(-1/m₂)| = 1/|m₂|.