If P G , the normal at P to a parabola y 2 = 4 a x cuts the axis in G and is produced to Q , so that G Q is equal to 1 2 P G ; prove that the other normals which pass through Q intersect at right angles.

Let the equation of the parabola be, y 2 = 4 a x And, let the coordinates of P be ( a m 1 2 , - 2 a m 1 ) . Here, t = - m 1 We know that equation of normal to parabola in parametric form is given by, y = - t x + 2 a t + a t 3 So, the equation of the normal at P can be given as, y = m 1 x - 2 a m 1 - a m 1 3 . . . ( i ) ( ∵ t = - m 1 ) Solving, equation ( i ) with the x -axis, y = 0 0 = m 1 x - 2 a m 1 - a m 1 3 ⇒ x = 2 a + a m 1 2 So, the coordinates of G are ( 2 a + a m 1 2 , 0 ) . The point Q is such that G Q = 1 2 P G ⇒ P Q = P G + G Q = 3 2 P G ⇒ P Q G Q = 3 2 P G 1 2 P G = 3 1 Hence, Q divides P G in the ratio of 3 : 1 , externally. Let the coordinates of Q be h , k . Then, we get, k = 3 × 0 - 1 - 2 a m 1 3 - 1 (Concept used - section formula for external division) k = a m 1 m 1 = k a . . . ( ii ) But, we know that, Equation of normal of a parabola in parametric form is y = - t x + 2 a t + a t 3 As t = - m ⇒ y = m x - 2 a m - a m 3 If it is passing through ( h , k ) , then it must satisfy it ∴ a m 3 + ( 2 a - h ) m + k = 0 ⇒ m 1 m 2 m 3 = - k a Substituting the value of m 1 in equation ( ii ) , we get, k a m 2 m 3 = - k a ⇒ m 2 m 3 = - 1 Hence, the two normals through Q are perpendicular to each other.

Prove that the equation to the circle, which passes through the focus and touches the parabola y 2 = 4 a x a > 0 at the point ( a t 2 , 2 a t ) is x 2 + y 2 - a x ( 3 t 2 + 1 ) - a y ( 3 t - t 3 ) + 3 a 2 t 2 = 0 . Also prove that the locus of its centre is the curve 27 a y 2 = 2 x - a x - 5 a 2 .

If the circle touches the parabola y 2 = 4 a x at ( a t 2 , 2 a t ) , then they must have a common tangent at that point, and hence a common normal. The centre of the circle must lie on that normal. Let ( h , k ) be the coordinates of the centre and r be the radius of the circle. Hence, its equation can be written as x 2 + y 2 - 2 h x - 2 k y + c = 0 . . . ( i ) The equation to the normal at ( a t 2 , 2 a t ) is y = - t x + 2 a t + a t 3 . . . ( ii ) As the centre ( h , k ) lies on the normal ( ii ) . Hence, k = - t h + 2 a t + a t 3 . . . ( iii ) Focus of the parabola is ( a , 0 ) . As the circle passes through ( a , 0 ) and ( a t 2 , 2 a t ) distance of these points from the centre ( h , k ) must be equal to the radius. ⇒ r 2 = h - a 2 + k 2 = h - a t 2 2 + k - 2 a t 2 ⇒ - 2 a h + a 2 = - 2 a h t 2 - 4 a k t + a 2 t 4 + 4 a 2 t 2 Dividing by a ⇒ 4 k t = 2 h ( 1 - t 2 ) + a t 4 + 4 a t 2 - a . . . ( iv ) Solving ( iii ) and ( iv ) , 4 t ( - t h + 2 a t + a t 3 ) = 2 h ( 1 - t 2 ) + a t 4 + 4 a t 2 - a ⇒ - 4 h t 2 + 8 a t 2 + 4 a t 4 = a t 4 - 2 h t 2 + 4 a t 2 + 2 h - a ⇒ 3 a t 4 - 2 h t 2 + 4 a t 2 - 2 h + a = 0 ⇒ a ( 3 t 4 + 4 t 2 + 1 ) = 2 h ( 1 + t 2 ) ⇒ ( 3 t 4 + 3 t 2 + t 2 + 1 ) = 2 h ( 1 + t 2 ) ⇒ a ( 3 t 2 + 1 ) ( 1 + t 2 ) = 2 h ( 1 + t 2 ) 2 h = a ( 3 t 2 + 1 ) . . . ( v ) and 2 k = a ( 3 t - t 3 ) . . . ( vi ) Putting the values of h and k in ( i ) , Equation of the circle is x 2 + y 2 - a x ( 3 t 2 + 1 ) - a y ( 3 t - t 3 ) y + c = 0 . . . ( vii ) As ( vii ) passes through the focus ( a , 0 ) , the coordinates will satisfy it. Hence, a 2 - a 2 ( 3 t 2 + 1 ) + c = 0 ⇒ c = 3 a 2 t 2 . Putting the values of c in ( vii ) , the circle is x 2 + y 2 - a x 3 t 2 + 1 - a y 3 t - t 3 + 3 a 2 t 2 = 0 Hence proved. Now, to get the locus of centre, we have to eliminate t from ( v ) and ( vi ) . Multiplying ( v ) by t and ( vi ) by 3 and adding them, 2 h t = a t ( 3 t 2 + 1 ) ⇒ 6 k = 3 a ( 3 t - t 3 ) ⇒ 2 t h + 6 k = 10 a t t = 3 k 5 a - h Using the equation ( v ) , 2 h = a ( 3 t 2 + 1 ) ⇒ 2 h - a = 3 a 3 k 5 a - h 2 ⇒ 2 h - a 5 a - h 2 = 27 a k 2 Simplifying and generalizing, the locus of centre ( h , k ) is 2 x - a x - 5 a 2 = 27 a y 2

HARD

JEE Advanced

IMPORTANT

Earn 100

Find the locus of a point O when the three normals drawn from the parabola $y^{2} = 4 a x$ . Prove that the normals at the points, where the straight line $l x + m y = 1$ meets the parabola, meet on the normal at the point $(\frac{4 a m^{2}}{l^{2}}, \frac{4 a m}{l})$ of the parabola.

Important Questions on The Parabola (Continued)

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 9 - The Parabola (Continued)>EXAMPLES XXX>Q 19

$P, Q, R, P', Q' and R'$ are points on the parabola $y^{2} = 4 a x$ . If the normals at the three points $P$ , $Q$ and $R$ meet in a point and if $P P'$ , $Q Q'$ and $R R'$ be chords parallel to $Q R$ , $R P$ and $t_{3} = - (t_{1} + t_{2}) = \frac{- 2 m}{l}$ , respectively, prove that the normals at $P'$ , $Q'$ and $R'$ also meet in a point.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 9 - The Parabola (Continued)>EXAMPLES XXX>Q 20

[If the normals drawn from any point to the parabola

y^{2} = 4 a x

cuts the line

x = 2 a

3

points whose ordinates are in arithmetical progression, prove that the tangents of the angles which the normals make with the axis are in geometrical progression.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 9 - The Parabola (Continued)>EXAMPLES XXX>Q 21

P G

, the normal at

P

to a parabola

y^{2} = 4 a x

cuts the axis in

G

and is produced to

Q

, so that

G Q

is equal to

\frac{1}{2} P G

; prove that the other normals which pass through

Q

intersect at right angles.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 9 - The Parabola (Continued)>EXAMPLES XXX>Q 22

Prove that the equation to the circle, which passes through the focus and touches the parabola

y^{2} = 4 a x (a > 0)

at the point

(a t^{2}, 2 a t)

x^{2} + y^{2} - a x (3 t^{2} + 1) - a y (3 t - t^{3}) + 3 a^{2} t^{2} = 0

. Also prove that the locus of its centre is the curve

27 a y^{2} = (2 x - a) {(x - 5 a)}^{2}

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 9 - The Parabola (Continued)>EXAMPLES XXX>Q 23

Show that three circles can be drawn to touch a parabola and also to touch at the focus of a given straight line passing through the focus. Also prove that the tangents at the point of contact with the parabola form an equilateral triangle.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 9 - The Parabola (Continued)>EXAMPLES XXX>Q 24

Through a point

P

, two tangents

P Q

and

P R

are drawn to a parabola

y^{2} = 4 a x (a > 0)

and circles are drawn through the focus to touch the parabola in

Q

and

R

, respectively. Prove that the common chord of these circles passes through the centroid of the triangle

P Q R

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 9 - The Parabola (Continued)>EXAMPLES XXX>Q 25

Prove that the locus of the centre of the circle, which passes through the vertex of a parabola

y^{2} = 4 a x (a > 0)

and ends of a normal chord of the parabola, is a parabola

2 y^{2} = a x - a^{2}

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 9 - The Parabola (Continued)>EXAMPLES XXX>Q 26

A circle is described whose Centre is the vertex and whose diameter is three-quarters of the length latus rectum of a parabola

y^{2} = 4 a x (a > 0)

. Prove that the common chord of the circle and parabola bisects the distance between the vertex and the focus.

Find the locus of a point O when the three normals drawn from the parabola y2=4ax. Prove that the normals at the points, where the straight line lx+my=1 meets the parabola, meet on the normal at the point 4am2l2,4aml of the parabola.

Important Questions on The Parabola (Continued)

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Find the locus of a point O when the three normals drawn from the parabola $y^{2} = 4 a x$ . Prove that the normals at the points, where the straight line $l x + m y = 1$ meets the parabola, meet on the normal at the point $(\frac{4 a m^{2}}{l^{2}}, \frac{4 a m}{l})$ of the parabola.