A variable line ax + by + c = 0,where a, b, c | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given the line $ax + by + c = 0$ where $a, b, c$ are in $A.P.$,we have $a + c = 2b$,which implies $a - 2b + c = 0$.This can be written as $a(1) +

Vedclass · Accepted Answer

$7$

Vedclass · Answer

(D) Given the line $ax + by + c = 0$ where $a, b, c$ are in $A.P.$,we have $a + c = 2b$,which implies $a - 2b + c = 0$.This can be written as $a(1) + b(-2) + c(1) = 0$,meaning the line always passes through the fixed point $(1, -2)$.Since the line is normal to the circle $(x - \alpha)^2 + (y - \beta)^2 = \gamma$,the center of the circle $(\alpha, \beta)$ must be $(1, -2)$.Thus,the circle equation is $(x - 1)^2 + (y + 2)^2 = \gamma$,which simplifies to $x^2 + y^2 - 2x + 4y + 5 - \gamma = 0$.This circle is orthogonal to $x^2 + y^2 - 4x - 4y - 1 = 0$.The condition for orthogonality $2g_1g_2 + 2f_1f_2 = c_1 + c_2$ gives:$2(-1)(-2) + 2(2)(-2) = (5 - \gamma) + (-1)$.$4 - 8 = 4 - \gamma$.$-4 = 4 - \gamma \Rightarrow \gamma = 8$.Finally,$\alpha + \beta + \gamma = 1 + (-2) + 8 = 7$.

$A$ variable line $ax + by + c = 0$,where $a, b, c$ are in $A.P.$,is normal to a circle $(x - \alpha)^2 + (y - \beta)^2 = \gamma$,which is orthogonal to the circle $x^2 + y^2 - 4x - 4y - 1 = 0$. The value of $\alpha + \beta + \gamma$ is equal to

Similar Questions

The centre$(s)$ of the circle$(s)$ passing through the points $(0, 0)$ and $(1, 0)$ and touching the circle $x^2 + y^2 = 9$ is/are:

Find the equation of the pair of tangents drawn from the origin to the circle $x^2 + y^2 + 20(x + y) + 20 = 0$.

The slopes of the focal chords of the parabola $y^2=32x$,which are tangents to the circle $x^2+y^2=4$,are

The values of the constant term in the equation of a circle passing through $(1, 2)$ and $(3, 4)$ and touching the line $3x + y - 3 = 0$ are

The locus of the midpoints of the chords of the circle $x^2 + y^2 + 4x - 6y - 12 = 0$ which subtend an angle of $\frac{\pi}{3}$ radians at its circumference is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module