If the line $lx + my + n = 0$ intersects the curve $a{x^2} + 2hxy + b{y^2} = 1$ at $P$ and $Q$ such that the circle with $PQ$ as a diameter passes through the origin, then ${l^2} + {m^2} = $

${n^2}{\left( {a + b} \right)^2}$

${n^2}\left( {{a^2} - {b^2}} \right)$

${n^2}\left( {{a^2} + {b^2}} \right)$

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JEE • MATHEMATICS • PAIR OF STRAIGHT LINES

Pair of straight lines practice · Hard

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1 Hard

1 of 4

Application +4 | -1

The equation of the pair of straight lines, each of which makes angle

\alpha

with the line

y=x

, is

View answer and explanation

Correct option: D

x^2-2 x y \sec 2 \alpha+y^2=0

Option D (

x^2-2 x y \sec 2 \alpha+y^2=0

) is correct because it represents the equation of the pair of straight lines, each of which makes an angle

\alpha

with the line

y=x

2 Hard

2 of 4

Analyzing +4 | -1

If the line

lx + my + n = 0

intersects the curve

a{x^2} + 2hxy + b{y^2} = 1

P

and

Q

such that the circle with

PQ

as a diameter passes through the origin, then

{l^2} + {m^2} =

View answer and explanation

Correct option: A

{n^2}\left( {a + b} \right)

Option A

{n^2}\left( {a + b} \right)

is correct because it is derived from the condition that the lines joining the origin to the intersection points are at right angles.

3 Hard

3 of 4

Analyzing +4 | -1

a x^3+b y^3+c x^2 y+d x y^2=0

represents three distinct straight lines, such that each bisects the angle between the other two, then which of the following is true?

View answer and explanation

Correct option: A

3 b+c=0

Option B (

3 a+d=0

) is correct because it represents the condition for the three distinct lines to bisect the angles between each other, derived from the relationship between the roots and coefficients of the cubic equation.

4 Hard

4 of 4

Application +4 | -1

The square of distance between the point of intersection of the lines represented by the equation

a x^2+2 h x y+b y^2+2 g x+2 f y+c=0

and origin, is

View answer and explanation

Correct option: A

\frac{c(a+b)-f^2-g^2}{a b-h^2}

Option A (

\frac{c(a+b)-f^2-g^2}{a b-h^2}

) is correct because it represents the square of the distance between the point of intersection of the lines and the origin.

Revision notes and key takeaways

Review PAIR OF STRAIGHT LINES in your textbook or class notes, then use this set to check your understanding.

Key Takeaways

✓ This set contains 4 single-correct-answer MCQs on PAIR OF STRAIGHT LINES.
✓ Try each question before opening its answer. Read the explanation where one is available.
✓ Sign in to save submitted answers. Revisit incorrect answers alongside your textbook or class notes.

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Pair of straight lines practice · Hard

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