Question

Assertion :Tangents are drawn from the point $(17,7)$ to the circle $x^2+y^2=169.$ Statement-$1$ 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. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Reason is true, because equation of any $2$ tangent to the circle $x^2+y^2=169$ is
$y=mx\pm \left(13\right)\sqrt{1+m^2}$

if it passes through $(h,k)$ then

$k=mh\pm 13\sqrt{1+m^2}$
Squaring we get

$(k-mh{)}^2=169(1+m^2)\Rightarrow (169-h^2)m^2+2mhk+(169-k^2)=0$

giving us the slopes of the two tangents to the circle from the point $(h,k)$

If these tangent are perpendicular then

$\frac{169-k^2}{169-h^2}=1\Rightarrow h^2+k^2=338$
and the locus of $(h,k)$ is $x^2+y^2=338$

Assertion is true as the point $(17,7)$ lies on the circle.