Question

Assertion :Tangents are drawn from the point $(17,7)$ to the circle $x^2+y^2=169$.
STATEMENT-$I$ : The tangents are mutually perpendicular Reason: STATEMENT-2: The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is $x^2+y^2=338$

• A. Statement -1 is True, Statement -2 is true; Statement-2 is a correct explanation for Statement-1
• B. Statement -1 is True, Statement -2 is true; Statement-2 is NOT a correct explanation for Statement-1
• C. Statement -1 is True, Statement -2 is False
• D. Statement -1 is False, Statement -2 is True

Answer 1

The correct option is A Statement -1 is True, Statement -2 is true; Statement-2 is a correct explanation for Statement-1

Let the equation of the tangent be $y=mx+c$

Then it passes through $17,7$
Hence

$7=17m+c$
$c=7-17m$

Hence the equation of the line will be

$y=mx+7-17m$

Now the equation of the circle is $x^2+y^2=169$

Substitution of the equation of the line in the circle yields

$x^2+\left(m\right(x-17)+7{)}^2=169$
$x^2+m^2(x-17{)}^2+14m(x-17)+49=169$
$x^2+m^2(x-17{)}^2+14m(x-17)=120$
$x^2(1+m^2)-34m^{2}x+14mx+289m^2-238m=120$

$x^2(1+m^2)+x(14m-34m^2)+289m^2-238m-120=0$

$b^2-4ac$

$=m^2(14-34m{)}^2-4(1+m^2\left)\right(289m^2-238m-120)$

$=4\left[m^2\right(7-17m{)}^2-(1+m^2)(289m^2-238m-120)]$

$=4\left[m^2\right(289m^2-238m+49)-(1+m^2\left)\right(289m^2-238m-120\left)\right]$

$=4[289m^4-238m^3+49m^2-289m^4+238m^3+120m^2-289m^2+238m+120]$

$=4[169m^2-289m^2+238m+120]$

$=4[-120m^2+238m+120]$

$=4(m-\frac{12}{5})(m+\frac{5}{12})$

Hence

$b^2-4ac=0$ implies

$m_1=\frac{12}{5}$ and $m_2=\frac{-5}{12}$

Hence

$m_1.m_2=-1$

Thus the tangents are mutually perpendicular.

Therefore the equation of the tangents will be

$y=\frac{12}{5}x-\frac{169}{5}$

$12x-5y=169$ ...(i)

$y=\frac{-5}{12}x+\frac{169}{12}$

$5x+12=169$ ...(ii)

Now if tangents are drawn from the circle of radius $2R$ to a circle of radius $R$ such that the circles are concentric then the tangents are mutually perpendicular.

Hence correct option is $A$