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Maths-

General

Easy

Question

If length of focal chord of y² = 4ax is λ, then angle between axis of parabola and focal chord is

± sin^–1 $\sqrt{\frac{2 a}{l}}$
± sin^–1 $\sqrt{\frac{4 a}{l}}$
± tan^–1 $\sqrt{\frac{4 a}{l}}$
None of these

The correct answer is: ± sin^–1 $square root of fraction numerator 4 a over denominator l end fraction end root$

Let any point on focal chord is P(h, k)

where h = a + r cos $proportional to$
k = r sin $proportional to$
$therefore$ Put (h, k) in y² = 4ax
$rightwards double arrow$ r² sin² = 4a (a + r cos $proportional to$ )
$therefore blank$ r₁ + r₂ = $fraction numerator 4 a cos invisible function application alpha over denominator sin to the power of 2 end exponent invisible function application alpha end fraction$ , r₁r₂ = $fraction numerator negative 4 a to the power of 2 end exponent over denominator sin to the power of 2 end exponent invisible function application alpha end fraction$
Now here length of focal chord is
$l$ = |r₁| + |r₂|
= r₁ –r₂
= $square root of left parenthesis r subscript 1 end subscript plus r subscript 2 end subscript right parenthesis to the power of 2 end exponent minus 4 r subscript 1 end subscript r subscript 2 end subscript end root$
$therefore$ $proportional to$ $l$ = 4a/sin² $proportional to$

Related Questions to study

Assertion: If a > 0 and b² – 4ac < 0 then the value of the integral $not stretchy integral fraction numerator d x over denominator a x to the power of 2 end exponent plus b x plus c end fraction$ will be of the type $mu$ tan^–1 $fraction numerator x plus A over denominator B end fraction$ + C, where A, B, C, $mu$ are constants.
Reason: If a > 0, b² – 4ac < 0 then ax² + b x + c can be written as sum of two squares.

Assertion: If a > 0 and b² – 4ac < 0 then the value of the integral $not stretchy integral fraction numerator d x over denominator a x to the power of 2 end exponent plus b x plus c end fraction$ will be of the type $mu$ tan^–1 $fraction numerator x plus A over denominator B end fraction$ + C, where A, B, C, $mu$ are constants.
Reason: If a > 0, b² – 4ac < 0 then ax² + b x + c can be written as sum of two squares.

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