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 Assertion is incorrect but Reason are correct

Answer 1

Hint : A tangent to a circle at point P with coordinates is a straight line that touches the circle at P of the equation, the tangent is perpendicular to the radius which joins the centre of the circle to the point P. Here, the locus of the points which are mutually perpendicular tangents can be drawn to the given circle is true as given the tangents are mutually perpendicular.

Complete step-by-step answer :
Reason is true, because equation of any 2 tangent to the circle ${x^2} + {y^2} = 169$ is
We have the slope form as $y = mx + c$ i.e., for circle, ${x^2} + {y^2} = {a^2}$ we have,
${x^2} + {y^2} = {13^2}$ , hence 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
$k = mh \pm \left( {13} \right)\sqrt {1 + {m^2}}$
Squaring the terms, we get:
${\left( {k - mh} \right)^2} = 169\left( {1 + {m^2}} \right)$
Simplifying the terms, we get:
$\Rightarrow$ $\left( {169 - {h^2}} \right){m^2} + 2mhk + \left( {169 - {k^2}} \right) = 0$
This, gives us the slopes of the two tangents to the circle from the point (h, k)
If these tangents are perpendicular then
$\dfrac{{169 - {k^2}}}{{169 - {h^2}}} = 1$
$\Rightarrow$ ${h^2} + {k^2} = 338$
The locus of (h, k) is ${x^2} + {y^2} = 338$
Assertion is true as the point (17,7) lies on the circle, hence option A is the right answer.

Note : We must note that the tangent to a circle is defined as a straight line which touches the circle at a single point. The point where the tangent touches a circle is known as the point of tangency or the point of contact, as given the tangents are mutually perpendicular, hence, both Assertion and Reason are correct and Reason is the correct explanation for Assertion.