Parabola

Q: The equation of the straight line(s) that touches both x² + y² = 2a² and y² = 8ax is /are
(A) y = x + 2a

(B) y = – x – 2a

(C) y = – x + 2a

(D) y = x – 2a

Sol. The equation of any tangent to the circle x² + y² = 2a² is

xcosθ + ysinθ = √2a . . . . . (1)

The equation of any tangent to the parabola y² = 8ax is

y = mx + 2a/m . . . . . (2)

Since (1) and (2) are identical

$\large \frac{cos\theta}{-m} = \frac{sin\theta}{1} = \frac{m}{\sqrt{2}}$

$\large cos\theta = \frac{-m^2}{\sqrt{2}} $ and $\large sin\theta = \frac{m}{\sqrt{2}} $

Squaring and adding, m⁴ + m² – 2 = 0

⇒ m² = 1 ⇒ m = ± 1

Substituting in (2) the equation of the required tangent is y = ± (x + 2a)

Hence (A) and (B) are the correct answers.

Q: If two chords drawn from the point (4 , 4) to the parabola x^2 = 4y are divided by line y = m x in the ratio 1 : 2 , then

(A) m < – √3

(B) m < –√3 – 1

(C) m > √3

(D) m > √3– 1

Sol. Point (4, 4) lies on the parabola.

Let the point of intersection of the line y = m x with the chords be (a , ma), then

$\large \alpha = \frac{4+2x_1}{3}$

$\large x_1 = \frac{3\alpha – 4}{2}$

and , $\large m \alpha = \frac{4 + 2 y_1}{3}$

$\large y_1 = \frac{3m \alpha – 4}{2}$

as (x₁, y₁) lies on the curve

$\large (\frac{3\alpha – 4}{2})^2 = 4 (\frac{3m\alpha – 4}{2}) $

⇒ 9α² + 16 – 24a = 8(3mα – 4)

Þ 9α² – 24α(1 + m) + 48 = 0

Þ 3α² – 8α(1 + m) + 16 = 0

two distinct chords are obtained

D > 0.

(8(1 + m))² – 4 . 3 . 16 > 0

⇒ (1 + m)² – 3 > 0

1 + m > √3 or 1 + m < –√3

⇒ m > √3– 1 or m < –( √3 + 1).

Hence (B), (C) and (D) are the correct answers.

Q: If normal chord at the point (t² , 2t) of the parabola y² = 4x subtends a right angle at the vertex then the value of t is/are

(A) √3

(B) √2

(C) -√3

(D) -√2

Sol. Normal at t t₁ = -t -2/t …(1)

AB ⊥ BC

⇒ $\large \frac{2t – 0}{t^2 – 0} . \frac{2t_1 – 0}{t_1^2 – 0} = -1 $

-4tt₁ = (tt₁)²

t₁ t = -4

⇒ t² + 2 = 4

⇒ t = ± √2

Hence (B) and (D) are the correct answers.

Q: Let S be the set of all possible values of parameter ‘a’ for which the points of intersection of the parabolas y² = 3ax and  $y = \frac{1}{2}(x^2 + ax + 5)$ y = are concyclic . Then S contains the interval(s)

(A) (– ∞ , – 2)

(B) (– 2, 0)

(C) (0, 2)

(D) (2, ∞)

Q: The equation 2y² + 3y – 4x – 3 = 0 represents a parabola for which

(A) length of latus rectum is 4

(B) equation of the axis is 4y + 3 = 0

(C) equation of directrix is 2x + 1 = 0

(D) equation of tangent at vertex is x = -33/32

Q:The co-ordinates of the point on the parabola y² = 8x , which is at minimum distance from the circle x² + (y + 6)² = 1 are :

(A) (2, -4)

(B) (18, -12)

(C) (2,4)

(D) None of these

Sol:

Equation of normal for parabola y² = 8x at Q(2m² , –4m) is

y = mx – 4m – 2m³ ….(1)

Now for minimum distance (1) will be common normal for both. So (1) must pass through centre C(0, –6).

⇒ m³ + 2m – 3 = 0

⇒ (m –1)(m² + m +3) = 0

⇒ m = 1

so required point is (2, –4)

Hence (A) is the correct answer.

Q: The circles on focal radii of a parabola as diameter touch

(A) the tangent at the vertex

(B) the axis

(C) the directrix

(D) none of these

Sol: Any point on parabola y² = 4ax with focus S (a, 0) is P(at² , 2at).

Centre of circle with PS as diameter is C $\large (\frac{a(1+t^2)}{2} , a t )$ and radius $\large = \frac{a}{2}(1+t^2)$

clearly the distance of centre C from tangent at vertex is equal to radius.

Hence (A) is the correct answer.

Q: If the chord y = mx + c subtends a right angle at the vertex of the parabola y² = 4ax, then the value of c is

(A) – 4am

(B) 4am

(C) –2am

(D) 2am

Q: If the normals at the end points of a variable chord PQ of the parabola y² – 4y – 2x = 0 are perpendicular, then the tangents at P and Q will intersect at

(A) x + y = 3

(B) 3x – 7 = 0

(C) y+3 = 0

(D) 2x + 5 = 0

Q: The circle drawn with variable chord x + ay – 5 = 0 (a being a parameter) of the parabola y² =20x as diameter will always touch the line

(A) x + 5 = 0

(B) y + 5 = 0

(C) x + y + 5 = 0

(D) x – y + 5 = 0