The point where the line 4x - 3y + 7 = 0 touc | vedclass

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

Question

(C) Let $(a, b)$ be the point where the line $4x - 3y + 7 = 0$ touches the circle $x^2 + y^2 - 6x + 4y - 12 = 0$.The equation of the tangent at $(a, b

Vedclass · Accepted Answer

$(-1, 1)$

Vedclass · Answer

(C) Let $(a, b)$ be the point where the line $4x - 3y + 7 = 0$ touches the circle $x^2 + y^2 - 6x + 4y - 12 = 0$.The equation of the tangent at $(a, b)$ to the circle is given by $xa + yb - 3(x + a) + 2(y + b) - 12 = 0$.Rearranging the terms,we get $(a - 3)x + (b + 2)y - (3a - 2b + 12) = 0$.Comparing this with the given line $4x - 3y + 7 = 0$,we have the ratio of coefficients:$\frac{a - 3}{4} = \frac{b + 2}{-3} = \frac{-(3a - 2b + 12)}{7} = k$.From this,$a = 4k + 3$ and $b = -3k - 2$.Substituting these into the third ratio: $-(3(4k + 3) - 2(-3k - 2) + 12) = 7k$.$-(12k + 9 + 6k + 4 + 12) = 7k$ $\Rightarrow -(18k + 25) = 7k$ $\Rightarrow -25k = 25$ $\Rightarrow k = -1$.Substituting $k = -1$ back into the expressions for $a$ and $b$:$a = 4(-1) + 3 = -1$ and $b = -3(-1) - 2 = 1$.Thus,the point of contact is $(-1, 1)$.

The point where the line $4x - 3y + 7 = 0$ touches the circle $x^2 + y^2 - 6x + 4y - 12 = 0$ is

Similar Questions

The equation of a line parallel to the line $3x + 4y = 0$ and touching the circle $x^{2} + y^{2} = 9$ in the first quadrant is:

If the slope of the tangent of the circle $S \equiv x^2+y^2-13=0$ at $(2,3)$ is $m$,then the point $\left(m, \frac{-1}{m}\right)$ is

The tangent to the circle $x^2 + y^2 = 5$ at the point $(1, -2)$ also touches the circle $x^2 + y^2 - 8x + 6y + 20 = 0$. Find its point of contact.

Suppose the angle between the tangents drawn from $(0,0)$ to the circle $(x+\lambda)^2+(y+1)^2=\lambda^2$ is $\frac{\pi}{2}$. Then,$\lambda$ satisfies

If a circle,whose centre is $(-1, 1)$,touches the straight line $x + 2y + 12 = 0$,then the coordinates of the point of contact are

Vedclass Test Series

Exam Paper Generator

Online Exam Module