Question

If the circles ${x^2} + {y^2} + 2ax + cy + a = 0$ and ${x^2} + {y^2} - 3ax + dy - 1 = 0$ intersect in two distinct points P and Q then the line 5x + by –a = 0 passes through P and Q for:
A. exactly one value of a
B. no value of a
C. infinite value of a
D. exactly two values of a.

Answer 1

Hint -In this question, we will first see the definition of the common chord of two circles. After this, we will find the equation of the common chord of the two circles. After finding the common chord we will compare it with the given equation of the line to get an expression in ‘a’.

Complete step-by-step answer:

Let see the definition of the common chord of two circles.

The line joining the points of intersection of two circles is called the common chord of the two circles.

If the equations of the circles are

${S_1} = {x^2} + {y^2} + 2{g_1}x + 2{f_1}y + {c_1} = 0$ and

${S_2} = {x^2} + {y^2} + 2{g_2}x + 2{f_2}y + {c_2} = 0$ , then the equation of the common chord is ${S_1} - {S_2} = 0$ .

Hence the equation of the common chord is $2x({g_1} - {g_2}) + 2y({f_1} - {f_2}) + {c_1} - {c_2} = 0$

The given equation the circles are

${x^2} + {y^2} + 2ax + cy + a = 0$ and ${x^2} + {y^2} - 3ax + dy - 1 = 0$ and is given that both are intersecting in two points.

Therefore, we can write the equation of the common chord as:

${S_1} - {S_2} = 0$.

Putting the values of ${S_1}$ and${S_2}$ in above equation, we get:

$({x^2} + {y^2} + 2ax + cy + a) - ({x^2} + {y^2} - 3ax + dy - 1) = 0$

$ \Rightarrow $ 5ax + (c – d)y + a +1=0.

So, the calculated equation of common chord of the given two circles is

5ax + (c – d)y + a +1=0. (1)

And the given equation of line is

5x + by –a = 0. (2)

For the given line to be the common chord of the given two circles. It must be equal to the equation of the common chord given by equation 1.

On equating the coefficients of two equations, we get:

$\dfrac{{5a}}{5} = \dfrac{{c - d}}{b} = \dfrac{{a + 1}}{{ - a}}$

$ \Rightarrow {a^2} + a + 1 = 0$ .

For the above equation, there is no real solution as D<0.

So, there is no value of ‘a’ for which the given line is the common chord of the two circles.

So, option B is correct.

Note- In this type of question, you must know to find the equation of the common chord. To find the equation of common chord, the coefficients of ${x^2}$and${y^2}$ in both the equations must be the same. Two circles touch each other if the length of their common chord is zero. The maximum length of the common chord is equal to the diameter of the smaller circle.