Pls answer 84 question
Q.84. If equation of one tangent drawn from (0, 0) to the circle with centre (2, 4) is 4x + 3y = 0, then equation of the other tangent from (0, 0) is
(1) 4x - 3y = 0
(2) x = 0
(3) y = 0
(4) x + 4y = 0

Dear Student,



$$\begin{gathered} Given : equation of one \tan gent from (0, 0) to the circle with centre (2, 4) is 4x+3y=0. \\ To Calculate : equation of the other \tan gent from (0,0). \\ solution : radius of the circle will be length of perpendicular on line 4x+3y=0 from centre (2, 4). \\ As We know, \\ length of perpendicular from a point (x_1, y_1) on the line ax+by+c=0 \\ is = \left|\frac{a{x}_1+b{y}_1+c}{\sqrt{a^2+b^2}}\right| \\ \Rightarrow Radius of the circle = \left|\frac{4\left(2\right)+3\left(4\right)+0}{\sqrt{4^2+3^2}}\right|=\left|\frac{8+12}{\sqrt{16+9}}\right|=\left|\frac{20}{5}\right|=4 units \\ So, equation of circle will be \\ (x-2{)}^2+(y-4{)}^2=4^2 \\ \Rightarrow x^2+2^2-2\times x\times 2+y^2+4^2-2\times y\times 4=16 \\ \Rightarrow x^2+y^2-4x-8y+20-16=0 \\ S : x^2+y^2-4x-8y+4=0 \\ g=\frac{coefficient of x}{2}=\frac{-4}{2}=-2 \\ f=\frac{coefficient of y}{2}=\frac{-8}{2}=-4 \\ equation of pair of \tan gents from a point P(x_1, y_1) is S{S}_1=T^2 \\ (x^2+y^2+2gx+2fy+c)({x_1}^2+{y_1}^2+2g{x}_1+2f{y}_1+c)=\left\{\right(x{x}_1+y{y}_1+g(x+x_1)+f(y+y_1)+c{\}}^2 \\ Here S : x^2+y^2-4x-8y+4=0 and P(x_1, y_1)=(0, 0) \\ \Rightarrow (x^2+y^2-4x-8y+4)( 0^2+0^2-4\times 0-8\times 0+4)=(x\times 0+y\times 0+(-2\left)\right(x+0)+(-4\left)\right(y+0)+4{)}^2 \\ \Rightarrow 4(x^2+y^2-4x-8y+4)=(-2x-4y+4{)}^2 \\ \Rightarrow 4x^2+4y^2-16x-32y+16=4x^2+16y^2+16+2\left\{\right(-2x\left)\right(-4y)+(-2x)4+(-4y\left)4\right\} \\ \Rightarrow 4x^2+4y^2-16x-32y+16=4x^2+16y^2+16+16xy-16x-32y \\ \Rightarrow \cancel{4x^2}+4y^2\cancel{-16x}\cancel{-32y}+\cancel{16}\cancel{-4x^2}-16y^2-\cancel{16}-16xy\cancel{+16x}\cancel{+32y}=0 \\ \Rightarrow -12y^2-16xy=0 \\ \Rightarrow -4y(3y+4x)=0 \\ \Rightarrow (4x+3y)y=0 \\ \sin ce one of the \tan gent is 4x+3y=0 \\ so other \tan gent will be y=0 or x axis. \end{gathered}$$

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