Question

Tangents are drawn from the point $(17,7)$ to the circle $x^2+y^2=169.$

Assertion: The tangents are mutually perpendicular. because

Reason: The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is $x^2+y^2=338.$

• A. Assertion is true, reason is true; reason is a correct explanation for assertion
• B. Assertion is true, reason is true; reason is Not a correct explanation for assertion
• C. Assertion is true, reason is false
• D. Assertion is false, reason is true

Answer 1

The correct option is A Assertion is true, reason is true; reason is a correct explanation for assertion

$x^2+y^2=169$

Equation of a pair of tangents from $(h,k)$ to the given circle is ${SS}_1=T^2$

$\Rightarrow {(hx+ky-169)}^2=(h^2+k^2-169)(h^2+k^2-169)$

$=h^2.x^2+k^2{y}^2+{\left(169\right)}^2+2hxy-338(hx+ky)$

$=x^2(h^2+k^2-169)+y^2(h^2+k^2-169)-169(h^2+k^2-169)$

$\therefore x^2(k^2-169)+y^2(h^2-169)-2hxy+338(hx+ky)-169(h^2-k^2)=0$

For the pair of lines to be mutually perpendiculars,

Coefficient of $x^2+$ coefficients of $y^2=0$

$\Rightarrow h^2+k^2-338=0\Rightarrow h^2+k^2=338$

Which is the equation of a director circle point $(17,7)$ satisfies the above equation

Hence Assertion and season are true, season is a cosset explanation of assertion.