If the circles x2 + y2 + 5Kx + 2y + K = 0 and 2(x2 + y2) + 2Kx + 3y -1 = 0, (K ∈ R) , intersect at the points P and Q, then the line 4x + 5y - K = 0 passes through P and Q for :

Question

If the circles x2 + y2 + 5Kx + 2y + K = 0 and 2(x2 + y2) + 2Kx + 3y -1 = 0, (K ∈ R) , intersect at the points P and Q, then the line 4x + 5y - K = 0 passes through P and Q for : (A) exactly two values of K (B) exactly one value of K (C) no value of K (D) infinitely many values of K

Tardigrade Admin · Accepted Answer

Equation of common chord 4kx + (1/2) y + k + (1/2) = 0 ....(1) and given line is 4x + 5y -k = 0 .....(2) On comparing (1) (2), we get k = (1/10) = ( k + (1/2)/-k) ⇒ No real value of k exist

Q. If the circles $x^{2} + y^{2} + 5 K x + 2 y + K = 0$ and $2 (x^{2} + y^{2}) + 2 K x + 3 y - 1 = 0, (K \in R)$ , intersect at the points $P$ and $Q$ , then the line $4 x + 5 y - K = 0$ passes through $P$ and $Q$ for :

exactly two values of K

exactly one value of K

no value of K

infinitely many values of K

Equation of common chord
$4 k x + \frac{1}{2} y + k + \frac{1}{2} = 0$ ....(1)
and given line is 4x + 5y -k = 0 .....(2)
On comparing (1) & (2), we get
$k = \frac{1}{10} = \frac{k + \frac{1}{2}}{- k}$
$\Rightarrow$ No real value of $k$ exist

Q. If the circles x2+y2+5Kx+2y+K=0 and 2(x2+y2)+2Kx+3y−1=0,(K∈R) , intersect at the points P and Q, then the line 4x+5y−K=0 passes through P and Q for :