Let $a$ and $b$ be two non-zero real numbers. The equation $(ax^2 + by^2 + c)(x^2 - 5xy + 6y^2) = 0$ represents:
- A$A$ circle and an ellipse,when $a$ and $b$ are of the same sign and $c$ is of sign opposite to that of $a$.
- BTwo straight lines and a hyperbola,when $a$ and $b$ are of the same sign and $c$ is of opposite sign to $a$.
- CFour straight lines,when $c = 0$ and $a, b$ are of the same sign.
- DTwo straight lines and a circle,when $a = b$ and $c$ is of sign opposite to that of $a$.
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For the circle $C$ with the equation $x^2+y^2-16x-12y+64=0$,match the List-$I$ with the List-$II$ given below.
List-$I$ List-$II$ $(i)$ The equation of the polar of $(-5, 1)$ with respect to $C$ $(A)$ $y = 0$ $(ii)$ The equation of the tangent at $(8, 0)$ to $C$ $(B)$ $y = 6$ $(iii)$ The equation of the normal at $(2, 6)$ to $C$ $(C)$ $x + y = 7$ $(iv)$ The equation of the diameter of $C$ through $(8, 12)$ $(D)$ $13x + 5y = 98$ $(E)$ $x = 8$
The correct match is:
| List-$I$ | List-$II$ |
| $(i)$ The equation of the polar of $(-5, 1)$ with respect to $C$ | $(A)$ $y = 0$ |
| $(ii)$ The equation of the tangent at $(8, 0)$ to $C$ | $(B)$ $y = 6$ |
| $(iii)$ The equation of the normal at $(2, 6)$ to $C$ | $(C)$ $x + y = 7$ |
| $(iv)$ The equation of the diameter of $C$ through $(8, 12)$ | $(D)$ $13x + 5y = 98$ |
| $(E)$ $x = 8$ |
The correct match is:
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