What is the locus of midpoint of chords of x2 + y2 + 2gx + 2fy + c = 0 that pass through the point (1, 1)?
What is the locus of midpoint of chords of x2 + y2 + 2gx + 2fy + c = 0 that pass through the point (1, 1)?
The correct option is C
We know how to find the equation of a chord with a given midpoint. In this question, the point given is not the midpoint. We will assume the point to be (h,k) like in
other locus problems. Once we get a relation between h and k, we will replace it with x and y.
We saw that T1 = S1 gives the equation of chord with a given midpoint.
When midpoint is (h, k) we can expand the above equation as,
hx + ky + g(x + h) + f(y + k) + c = h2 + k2 + 2gh + 2fk + c→ (1)
since this chord passes through (1, 1), we give (x, y)≡ (1, 1) in the above equation.
i.e., h + k + g (1 + h) + f(1 + k) = h2 + k2 + 2gh + 2fk
Rearranging this
h2 + k2 + h(g − 1) + k(f − 1) − (g + f) = 0
so this is the equation of the locus of all such points (h, k)
i.e., The required equation is,
x2 + y2 + x(g − 1) + y(f − 1) − (g + f) = 0
We know how to find the equation of a chord with a given midpoint. In this question, the point given is not the midpoint. We will assume the point to be (h,k) like in
other locus problems. Once we get a relation between h and k, we will replace it with x and y.
We saw that T1 = S1 gives the equation of chord with a given midpoint.
When midpoint is (h, k) we can expand the above equation as,
hx + ky + g(x + h) + f(y + k) + c = h2 + k2 + 2gh + 2fk + c→ (1)
since this chord passes through (1, 1), we give (x, y)≡ (1, 1) in the above equation.
i.e., h + k + g (1 + h) + f(1 + k) = h2 + k2 + 2gh + 2fk
Rearranging this
h2 + k2 + h(g − 1) + k(f − 1) − (g + f) = 0
so this is the equation of the locus of all such points (h, k)
i.e., The required equation is,
x2 + y2 + x(g − 1) + y(f − 1) − (g + f) = 0
