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Point Form of Tangent: Hyperbola

lf the area o...

Question

lf the area of the quadrilateral formed by the tangent from the origin to the circle $x^{2} + y^{2} + 6 x - 10 y + c = 0$ and the pair of radii at the points of contact of these tangents to the circle is $8$ square units, then $c$ is a root of the equation

c^{2} - 32 c + 64 = 0

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c^{2} - 34 c + 64 = 0

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c^{2} + 2 c - 64 = 0

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c^{2} + 34 c - 64 = 0

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Solution

The correct option is B $c^{2} - 34 c + 64 = 0$
$A C = A B =$ Length of tangent from $(0, 0)$ to
$S : x^{2} + y^{2} + 6 x - 10 y + c = 0$
$A C = A B = \sqrt{S_{1}} = \sqrt{c}$
Area of quadrilateral $A B O C =$ Area of $△ A B O +$ Area of $△ A C O$
$= \frac{1}{2} \times A B \times r + \frac{1}{2} \times A C \times r$

$= \frac{r}{2} [\sqrt{c} + \sqrt{c}]$
$= r \sqrt{c}$
Given : $r \sqrt{c} = 8$
$\sqrt{34 - c} \times \sqrt{c} = 8$
$\Rightarrow 34 c - c^{2} = 64$
$\Rightarrow c^{2} - 34 c + 64 = 0$

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Similar questions

Q. If the area of the quadrilateral formed by the tangent from the origin to the circle

x^{2} + y^{2} + 6 x - 10 y + c = 0

and the pair of radii at the points of contact of these tangents to tbe circle is

8

square units. then

c

is a root of the equation

Q. Let

C_{1}

and

C_{2}

are circles defined by

x^{2} + y^{2} - 20 x + 64 = 0

and

x^{2} + y^{2} + 30 x + 144 = 0

. The length of the shortest line segment PQ that is tangent to

C_{1}

at P and

C_{2}

at Q is

Q. If the area of the quadrilateral formed by the tangents from the origin to the circle

x^{2} + y^{2} + 6 x - 10 y + c = 0

and radii corresponding to the point of contact is

15

sq. units, then value(s) of

c

can be

If $c_{1} : x^{2} + y^{2} - 20 x + 64 = 0$ and $c_{2} :$ $x^{2} + y^{2} + 30 x + 144 = 0$ . Then the length of the shortest line segment $P Q$ which is tangent to $c_{1}$ at $P$ and $c_{2}$ at $Q$ is

Q. If

C_{1}

and

C_{2}

are circles whose equations are

x^{2} + y^{2} - 20 x + 64 = 0

and

x^{2} + y^{2} + 30 x + 144 = 0

, then the length of the shortest line segment

P Q

that is tangent to

C_{1}

P

and to

C_{2}

Q

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lf the area of the quadrilateral formed by the tangent from the origin to the circle x2+y2+6x−10y+c=0 and the pair of radii at the points of contact of these tangents to the circle is 8 square units, then c is a root of the equation

The correct option is B c2−34c+64=0AC=AB= Length of tangent from (0,0) toS:x2+y2+6x−10y+c=0AC=AB=√S1=√cArea of quadrilateral ABOC= Area of △ABO+ Area of △ACO=12×AB×r+12×AC×r=r2[√c+√c]=r√cGiven : r√c=8√34−c×√c=8⇒34c−c2=64⇒c2−34c+64=0

lf the area of the quadrilateral formed by the tangent from the origin to the circle $x^{2} + y^{2} + 6 x - 10 y + c = 0$ and the pair of radii at the points of contact of these tangents to the circle is $8$ square units, then $c$ is a root of the equation