Question

Assertion :If the point $(2,4)$ is interior to the circle $x^2+y^2-6x-10y+k=0$ and the circle does not cut the axes at any point, then $25<k<32$ Reason: If the point $(x_1,y_1)$ lies inside the circle $x^2+y^2+2gx+2fy+c=0$ then ${x_1}^2+{y_1}^2+2g{x}_1+2f{y}_1+c<0$.

• 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

Since the point $(2,4)$ is interior to the given circle,

$\therefore 2^2+4^2-6\times 2-10\times 4+k<0\Rightarrow k-32<0\Rightarrow k<32$ ....(1)

Solving $y=0,x^2+y^2-6x-10y+k=0$

we get $x^2-6x+k=0$

which must have imaginary roots

$\therefore$ Discriminant $=36-4k<0\Rightarrow k>9$ ...(2)

Again, solving $x=0,x^2+y^2-6x-10y+k=0$

we get $y^2-10y+k=0$

which must have imaginary roots

$\therefore$ discriminant $=100-4k<0\Rightarrow k>25$ ...(3)

From (1),(2) and (3) we get

$25<k<32$

Thus the reason is the correct explanation for the assertion.