**(I) Select the correct option from the given alternatives.**

**Question 1.**

**Question 2.The length of latus rectum of the parabola – 4x – 8y + 12 = 0 is ________**

(A) 4

(B) 6

(C) 8

(D) 10

**Answer:**

(C) 8

Hint:

Given equation of parabola is

**Question 3.If the focus of the parabola is (0, -3), its directrix is y = 3, then its equation is ________**

Hint:

**Question 4.The co-ordinates of a point on the parabola y ^{2} = 8x whose focal distance is 4 are ________**

(B) (1, ±2√2)

(C) (2, ±4)

(D) none of these

**Answer:**

(C) (2, ±4)

**Question 5.The end points of latus rectum of the parabola y ^{2} = 24x are ________**

(A) (6, ±12)

(B) (12, ±6)

(C) (6, ±6)

(D) none of these

**Answer:**

(A) (6, ±12)

**Question 6.Equation of the parabola with vertex at the origin and directrix with equation x + 8 = 0 is ________**

(A) = 8x

(B) = 32x

(C) = 16x

(D) = 32y

**Answer:**

(B) = 32x

Hint:

Since directrix is parallel to Y-axis,

The X-axis is the axis of the parabola.

Let the equation of parabola be = 4ax.

Equation of directrix is x + 8 = 0

∴ a = 8

∴ required equation of parabola is = 32x

**Question 7.The area of the triangle formed by the lines joining the vertex of the parabola x ^{2} = 12y to the endpoints of its latus rectum is ________**(A) 22 sq. units

(B) 20 sq. units

(C) 18 sq. units

(D) 14 sq. units

**Answer:**

(C) 18 sq. units

Hint:

= 12y

4b = 12

b = 3

**Question 8.**

**Question 9.The equation of the parabola having (2, 4) and (2, -4) as end points of its latus rectum is ________**

(A) = 4x

(B) = 8x

(C) = -16x

(D) = 8y

**Answer:**

(B) = 8x

Hint:

The given points lie in the 1st and 4th quadrants.

∴ Equation of the parabola is = 4ax

End points of latus rectum are (a, 2a) and (a, -2a)

∴ a = 2

∴ required equation of parabola is y = 8x

**Question 10.If the parabola = 4ax passes through (3, 2), then the length of its latus rectum is ________**

**Question 11.**

**Question 12.**

**Question 14.If the line 4x – 3y + k = 0 touches the ellipse 5 + 9 = 45, then the value of k is**

(A) 21

(B) ±3√21

(C) 3

(D) 3(21)

**Answer:**

(B) ±3√21

**Question 15.The equation of the ellipse is 16 + 25 = 400. The equations of the tangents making an angle of 180° with the major axis are**(A) x = 4

(B) y = ±4

(C) x = -4

(D) x = ±5

**Answer:**

(B) y = ±4

**Question 16.The equation of the tangent to the ellipse 4 + 9 = 36 which is perpendicular to 3x + 4y = 17 is**

(A) y = 4x + 6

(B) 3y + 4x = 6

(C) 3y = 4x + 6√5

(D) 3y = x + 25

**Answer:**

(C) 3y = 4x + 6√5

**Question 17.**

**Question 18.**

**Question 19.If the line 2x – y = 4 touches the hyperbola 4 – 3 = 24, the point of contact is**(A) (1, 2)

(B) (2, 3)

(C) (3, 2)

(D) (-2, -3)

**Answer:**

(C) (3, 2)

**Question 20.The foci of hyperbola 4 – 9 – 36 = 0 are**

(A) (±√13, 0)

(B) (±√11, 0)

(C) (±√12, 0)

(D) (0, ±√12)

**Answer:**

(A) (±√13, 0)

II. Answer the following.

**Question 1.For each of the following parabolas, find focus, equation of file directrix, length of the latus rectum and ends of the latus rectum.**

**Question 2.Find the cartesian co-ordinates of the points on the parabola = 12x whose parameters are**(i) 2

(ii) -3

Solution:

Given equation of the parabola is = 12x

Comparing this equation with = 4ax, we get

4a = 12

∴ a = 3

If t is the parameter of the point P on the parabola, then

P(t) = (a, 2at)

i.e., x = a and y = 2at …..(i)

(i) Given, t = 2

Substituting a = 3 and t = 2 in (i), we get

x = 3(2 and y = 2(3)(2)

x = 12 and y = 12

∴ The cartesian co-ordinates of the point on the parabola are (12, 12).

(ii) Given, t = -3

Substitùting a = 3 and t = -3 in (i), we get

x = 3(-3 and y = 2(3)(-3)

∴ x = 27 and y = -18

∴ The cartesian co-ordinates of the point on the parabola are (27, -18).

**Question 3.Find the co-ordinates of a point of the parabola = 8x having focal distance 10.Solution:**Given equation of the parabola is = 8x

Comparing this equation with = 4ax, we get

4a = 8

∴ a = 2

Focal distance of a point = x + a

Given, focal distance = 10

x + 2 = 10

∴ x = 8

Substituting x = 8 in = 8x, we get

= 8(8)

∴ y = ±8

∴ The co-ordinates of the points on the parabola are (8, 8) and (8, -8).

**Question 4.Find the equation of the tangent to the parabola = 9x at the point (4, -6) on it.Solution:**

Given equation of the parabola is = 9x

Comparing this equation with = 4ax, we get

4a = 9

**Question 5.Find the equation of the tangent to the parabola = 8x at t = 1 on it.Solution:**Given equation of the parabola is = 8x

Comparing this equation with = 4ax, we get

4a = 8

a = 2

t = 1

Equation of tangent with parameter t is yt = x + a

∴ The equation of tangent with t = 1 is

y(1) = x + 2(1

y = x + 2

∴ x – y + 2 = 0

**Question 6.Find the equations of the tangents to the parabola = 9x through the point (4, 10).Solution:**Given equation of the parabola is = 9x

Comparing this equation with = 4ax, we get

**Question 7.**

Alternate method:

Comparing the given equation with = 4ax, we get

4a = 24

⇒ a = 6

Equation of the directrix is x = -6.

The given point lies on the directrix.

Since tangents are drawn from a point on the directrix are perpendicular,

Tangents drawn to the parabola = 24x from the point (-6, 9) are at the right angle.

**Question 8.**

**Question 9.A line touches the circle + = 2 and the parabola = 8x. Show that its equation is y = ±(x + 2).Solution:**Given equation of the parabola is = 8x

Comparing this equation with = 4ax, we get

4a = 8

a = 2

**Question 10.Two tangents to the parabola = 8x meet the tangents at the vertex in P and Q. If PQ = 4, prove that the locus of the point of intersection of the two tangents is = 8(x + 2).Solution:**

Given parabola is = 8x

Comparing with = 4ax, we get,

4a = 8

⇒ a = 2

**Question 11.**

∴ SP subtends a right angle at Q.

**Question 13.Find the(i) lengths of the principal axes(ii) co-ordinates of the foci(iii) equations of directrices(iv) length of the latus rectum(v) Distance between foci(vi) distance between directrices of the curve**∴ a = 5 and b = 3

Since a > b,

X-axis is the major axis and Y-axis is the minor axis.

(i) Length of major axis = 2a = 2(5) = 10

Length of minor axis = 2b = 2(3) = 6

∴ Lengths of the principal axes are 10 and 6.

(ii) We know that

Co-ordinates of foci are S(ae, 0) and S'(-ae, 0),

i.e., S(4√2, 0) and S'(-4√2, 0)

**Question 14.**

**Question 15.Find the eccentricity of an ellipse if the distance between its directrices is three times the distance between its foci.Solution:**

**Question 16.**

i.e., S(5√5, 0) and S'(-5√5, 0)

Since S, A and S’ lie on the X-axis,

SA = |5√5 – 10| and S’A = |-5√5 – 10|

= |-(5√5 + 10)|

= |5√5 + 10|

∴ SA . S’A = |5√5 – 10| |5√5 + 10|

= |(5√5)

^{2}– (10)

^{2}|

= |125 – 100|

= |25|

SA . S’A = 25

**Question 17.**

Squaring both the sides, we get

4 + 8m + 4 = 5 + 4

∴ – 8m = 0

∴ m(m – 8) = 0

∴ m = 0 or m = 8

These are the slopes of the required tangents.

∴ By slope point form y – = m(x – ),

the equations of the tangents are

y + 2 = 0(x – 2) and y + 2 = 8(x – 2)

∴ y + 2 = 0 and y + 2 = 8x – 16

∴ y + 2 = 0 and 8x – y – 18 = 0.

**Question 18.Solution:**

**Question 20.**

∴ xy = – 5

∴ – xy – 5 = 0, which is the required equation of the locus of P.

**Question 21.**

p

_{2}= length of perpendicular segment from S'(-3, 0) to the tangent (i)

**Question 22.Find the equation of the hyperbola in the standard form if(i) Length of conjugate axis is 5 and distance between foci is 13.(ii) eccentricity is 2/3 and distance between foci is 12.(iii) length of the conjugate axis is 3 and the distance between the foci is 5.Solution:**

**Question 23.**

**Question 24.**

**Question 25.**

**Question 26.**