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Standard VII

Mathematics

Radii, Chord, Diameter and Circumference of the Circle

Consider L1:...

Question

Consider
$L_{1} : 2 x + 3 y + p - 3 = 0$
$L_{2} : 2 x + 3 y + p + 3 = 0$ ,
where $p$ is a real number, and
$C : x^{2} + y^{2} + 6 x - 10 y + 30 = 0$ .
STATEMENT 1 : If line $L_{1}$ is a chord of circle $C$ , then line $L_{2}$ is not always a diameter of circle $C$ .
STATEMENT 2 : If line $L_{1}$ is a diameter of circle $C$ , then line $L_{2}$ is not a chord of circle $C$ .

STATEMENT 1 is True, STATEMENT 2 is True; STATEMENT 2 is a correct explanation for STATEMENT 1.

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STATEMENT 1 is True, STATEMENT 2 is True; STATEMENT 2 is NOT a correct explanation for STATEMENT 1.

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STATEMENT 1 is True, STATEMENT 2 is False.

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STATEMENT 1 is False, STATEMENT 2 is True.

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Solution

The correct option is D STATEMENT 1 is True, STATEMENT 2 is False.
Circle, $C \equiv (x + 3)^{2} + (y - 5)^{2} = 4$
Center $\equiv (- 3, 5)$ and radius $= 2$
Distance between $L_{1}$ and $L_{2}$ , $d = ∣ ∣ ∣ \frac{p + 3 - (p - 3)}{2^{2} + 3^{2}} ∣ ∣ ∣$
$\Rightarrow \frac{6}{\sqrt{13}} <$ radius, since both the lines are parallel.
$\Rightarrow$ statement (2) is false
But statement (1) is correct.

Hence, option C.

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Similar questions

Q. Assertion :Consider

L_{1} : 2 x + 3 y + p - 3 = 0 L_{2} : 2 x + 3 y + p + 3 = 0,

where p is a real number, and

C : x^{2} + y^{2} + 6 x - 10 y + 30 = 0

Statement-1: If line

L_{1}

is a chord of circle

C

, then

L_{2}

is not always a diameter of circle

C .

Reason: If line

L_{1}

is a diameter of circle

C

, then

L_{2}

not a chord of circle

C

Q. Statement 1: If

P_{1} (x_{1}, y_{1})

and

P_{2} (x_{2}, y_{2})

be the images of point

P (x, y)

about lines

L_{1} \equiv a x + b y + c

and

L_{2} \equiv b x - a y + c^{'} = 0

respectively, then the line joining points

P_{1}

and

P_{2}

always passes through point of intersection of lines

L_{1}

and

L_{2}

Statement 2: Lines

L_{1}

and

L_{2}

are perpendicular

Q. Lines

L_{1} : y - x = 0

and

L_{2} : 2 x + y = 0

intersect the line

L_{3} : y + 2 = 0

P

and

Q

, respectively. The bisector of the acute angle between

L_{1}

and

L_{2}

intersects

L_{3}

at R.

STATEMENT-1 : The ratio

P R : R Q

equals

2 \sqrt{2} : \sqrt{5} .

because

STATEMENT-2: In any triangle, bisector of an angle divides the triangle into two similar triangles.

Q. Assertion :Consider

L_{1} : 2 x + 3 y + p - 3 = 0

L_{2} : 2 x + 3 y + p + 3 = 0.

where p is a real number and

C : x^{2} + y^{2} + 6 x - 10 y + 30 = 0

If the line

L_{1}

is a chord of the circle C then the line is

L_{2}

not always a diameter of the circle C and

Reason: If the line

L_{1}

is a diameter of the circle C then the line

L_{2}

is not a chord of the circle C.

Statement 1: If a line $L = 0$ is tangent to the circle $S = 0$ , then it will also be a tangent to the circle $S + λ L = 0$

Statement 2: If a line touches a circle, then perpendicular distance of the line from the centre of the circle is equal to the radius of the circle.

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ConsiderL1:2x+3y+p−3=0L2:2x+3y+p+3=0,where p is a real number, and C:x2+y2+6x−10y+30=0.STATEMENT 1 : If line L1 is a chord of circle C, then line L2 is not always a diameter of circle C.STATEMENT 2 : If line L1 is a diameter of circle C, then line L2 is not a chord of circle C.