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The correct answer is (B) x=-25

The best I can EXPLAIN: Comparing equation with y^2=4ax.

4a=100 => a=25. DIRECTRIX is a LINE parallel to latus rectum in such a way that vertex is at middle of both.

Equation of directrix is x=-a => x=-25.

The CORRECT answer is (c) 5 UNITS

The best explanation: Equation of circle with center at y-axis (0, B) and radius r units is

(x)^2+(y-b)^2=r^2

=>(4)^2+(-b)^2=r^2

And (0)^2+(2-b)^2=r^2

=>(b-2)^2=b^2+4^2 => (-2)(2b-2)=16 => b-1=-4 => b=-3

So, r^2=4^2+3^2=5^2 => r=5 units.

Right OPTION is (a) x^2+y^2=25

Explanation: EQUATION of CIRCLE with center at (a, b) and radius R units is

(x-a)^2+(y-b)^2=r^2

So, equation of circle is (x-0)^2+(y-0)^2=5^2 => x^2+y^2=25.

The CORRECT choice is (C) (25, 0)

The best explanation: Comparing EQUATION with y^2=4ax.

4a=100 => a=25.

Focus is at (a, 0) i.e. (25, 0).

Correct answer is (d) two, two

For EXPLANATION I WOULD SAY: A hyperbola has two vertices lying on each end and two foci lying inside the hyperbola.

If P is a point on hyperbola and F1 and F2 are foci then |PF1-PF2| remains CONSTANT.

The correct OPTION is (d) y=0

The explanation is: Axis is the line passing through focus which DIVIDES the parabola SYMMETRICALLY into two equal halves. EQUATION of axis of parabola y^2=24x is y=0.

The CORRECT option is (B) (0, 0)

Explanation: Vertex is close END point of the PARABOLA i.e. origin (0, 0) as it SATISFIES the equation (y-0)^2=4a(x-0) also.

Right choice is (B) x=-25

To explain I would say: Comparing equation with y^2=-4ax.

4a=100 => a=25. Line PASSING through focus perpendicular to axis is latus RECTUM.

Equation of latus rectum is x=-a => x=-25.

Right option is (c) 8 units

For EXPLANATION I would say: Comparing the EQUATION with \((\frac{X}{a})^2+(\frac{y}{b})^2\)=1, we get a=5 and b=4.

Minor axis length = 2b = 2*4 = 8 units.

Correct OPTION is (c) y^2=12x

For explanation I would say: EQUATION of parabola which is symmetric about x-axis with vertex (0, 0) is y^2=4ax

Since parabola PASS through (3, 6) then 6^2=4a*3 => a=3.

So, equation is y^2=12x.

Correct ANSWER is (C) y=-25

For explanation: Comparing equation with x^2=-4ay.

4a=100 => a=25. Line PASSING through focus perpendicular to axis is latus RECTUM.

Equation of latus rectum is y=-a => y=-25.

Right answer is (c) on the circle

Explanation: Circle has EQUATION x^2+y^2-2x-4y-16=0.

6^2+2^2-2*6-4*2-36 = 36+4-12-8-36 =-16<0 so, point is INSIDE the circle.

Right CHOICE is (b) False

To EXPLAIN: No, center and VERTEX are different for hyperbola.

Hyperbola has one center and TWO VERTICES.

Correct answer is (a) x=25

For explanation I WOULD SAY: Comparing EQUATION with y^2=-4ax.

4a=100 => a=25. Directrix is a line parallel to latus RECTUM in such a way that VERTEX is at middle of both.

Equation of directrix is x=a => x=25.

The correct CHOICE is (a) x^2+y^2+2xy-4x-16y+54=0

Explanation: We KNOW, if P is a point on parabola, M is FOOT of perpendicular DRAWN from point P to directrix of parabola and F is focus of parabola then PF=PM

(x-2)^2+(y-5)^2 = (|x+y-2|/√2)^2

x^2+y^2+2xy-4x-16y+54=0.

The correct ANSWER is (c) 40 units

Best explanation: COMPARING equation with y^2=4ax.

4A=40. LENGTH of latus rectum of parabola is 4a =40.

Right choice is (a) (0, 25)

For EXPLANATION I WOULD say: COMPARING equation with x^2=4ay.

4a=100 => a=25.

Focus is at (0, a) i.e. (0, 25).

Right ANSWER is (a) \((\frac{x}{4})^2-(\frac{y}{3})^2\)=1

Explanation: GIVEN, a=3 and c=5 => B^2=c^2-a^2 = 5^2-3^2=4^2 => b=4.

Equation of hyperbola with TRANSVERSE axis along y-axis is \((\frac{y}{a})^2-(\frac{x}{b})^2\)=1.

So, equation of given hyperbola is \((\frac{y}{3})^2-(\frac{x}{4})^2\)=1.

The CORRECT CHOICE is (b) x^2=12y

To explain I would say: EQUATION of parabola which is symmetric about y-axis with vertex (0, 0) is x^2=4ay

Since parabola pass through (6, 3) then 6^2=4a*3 => a=3.

So, equation is x^2=12y.

Correct answer is (a) (±3,0)

Best explanation: Comparing the equation with \((\FRAC{x}{a})^2+(\frac{y}{b})^2\)=1, we GET a=5 and b=4.

For ellipse, c^2=a^2-b^2=25-16=9 => c=3.

So, COORDINATES of foci are (±c,0) i.e. (±3,0).

The CORRECT option is (B) False

To ELABORATE: No, center and vertex are different for ellipse.

Ellipse has ONE center and two VERTICES.

Correct choice is (b) equilateral

For EXPLANATION I would SAY: A hyperbola in which length of transverse and CONJUGATE axis is EQUAL is called equilateral hyperbola. In this type of hyperbola, a=b i.e. 2a=2b or length of transverse and conjugate axis are equal.

Correct CHOICE is (a) x=±5

Best EXPLANATION: COMPARING the equation with \((\frac{x}{a})^2-(\frac{y}{b})^2\)=1, we get a=3 and b=4.

For HYPERBOLA, c^2=a^2+b^2= 9+16=25 => c=5.

Equation of LATUS rectum x=±c i.e. x= ±5.

The CORRECT option is (c) (0,±3)

To explain I would SAY: Comparing the equation with \((\frac{x}{b})^2+(\frac{y}{a})^2\)=1, we GET a=5 and b=4.

For ellipse, c^2=a^2-b^2=25-16=9 => c=3.

So, COORDINATES of foci are (0,±c) i.e. (0,±3).

Correct CHOICE is (a) x=±3

To ELABORATE: Comparing the equation with \((\frac{x}{a})^2+(\frac{y}{B})^2\)=1, we get a=5 and b=4.

For ELLIPSE, c^2=a^2-b^2=25-16=9 => c=3.

Equation of latus rectum x=±c i.e. x=±3.

Correct choice is (d) (-25, 0)

For explanation I would SAY: COMPARING equation with y^2=-4ax.

4a=100 => a=25.

Focus is at (-a, 0) i.e. (-25, 0).

Right ANSWER is (b) 3/5

To EXPLAIN: Comparing the equation with \((\frac{X}{a})^2+(\frac{y}{b})^2\)=1, we GET a=5 and b=4.

For ellipse, c^2=a^2-b^2= 25-16=9 => c=3.

We KNOW, for ellipse c=a*e

So, e=c/a = 3/5.

Right choice is (a) x=0

The best I can EXPLAIN: Axis is the line passing through focus which DIVIDES the PARABOLA symmetrically into TWO equal HALVES. Equation of axis of parabola x^2=24y is x=0.

The correct option is (C) (0,±5)

For explanation: Comparing the equation with \((\frac{y}{a})^2-(\frac{x}{B})^2\)=1, we get a=4 and b=3.

For hyperbola, c^2=a^2+b^2= 16+9=25 => c=5.

So, coordinates of FOCI are (0,±c) i.e. (0,±5).

The correct OPTION is (c) on the circle

The best I can EXPLAIN: Circle has equation x^2+y^2-2x-4y=0.

0^2+0^2-2*0-4*0+0 = 0 so, point is on the circle.

Right CHOICE is (b) x^2+y^2-4x-10y+4=0

Explanation: Equation of circle with center at (a, b) and radius R UNITS is

(x-a)^2 + (y-b)^2 = r^2

(5-2)^2 + (9-5)^2 = r^2 => r^2=3^2+4^2 => r=5.

So, equation of circle is (x-2)^2+(y-5)^2=5^2 => x^2+y^2-4x-10y+4=0.

Correct answer is (C) 5 units

To EXPLAIN: COMPARING the equation with general form x^2+y^2+2gx+2fy+c=0, we get

2g=-4 => g=-2

2f=-10 => f=-5

c=4

Radius = \(\SQRT{g^2+f^2-c} = \sqrt{4+25-4}\)=5.

The correct option is (B) x^2+y^2-4x-10y+4=0

The best I can explain: EQUATION of circle with center at (a, b) and RADIUS R UNITS is

(x-a)^2+(y-b)^2=r^2

So, equation of circle is (x-2)^2+(y-5)^2=5^2 => x^2+y^2-4x-10y+4=0.

Right ANSWER is (a) \((\frac{x}{4})^2-(\frac{y}{5})^2\)=1

Easiest explanation: Given, 2a=8 => a=4 and 2b=10 => b=5.

Equation of hyperbola with transverse AXIS ALONG x-axis is \((\frac{x}{4})^2-(\frac{y}{5})^2\)=1.

So, equation of given hyperbola is \((\frac{x}{4})^2-(\frac{y}{5})^2\)=1.

Right choice is (B) 32/3

For explanation I WOULD say: Comparing the equation with \((\frac{x}{a})^2-(\frac{y}{b})^2\)=1, we get a=3 and b=4.

We know, length of LATUS rectum = 2b^2/a.

So, length of latus rectum of given HYPERBOLA = 2*42/3 = 32/3.

Correct CHOICE is (d) 5/3

The best I can explain: Comparing the equation with \((\frac{X}{a})^2-(\frac{y}{B})^2\)=1, we GET a=3 and b=4.

For hyperbola, c^2=a^2+b^2= 9+16=25 => c=5.

We know, for hyperbola c=a*e

So, e=c/a = 5/3.

Right option is (a) (±5,0)

The BEST I can explain: Comparing the equation with \((\frac{x}{a})^2-(\frac{y}{B})^2\)=1, we GET a=3 and b=4.

For hyperbola, c^2=a^2+b^2=9+16=25 => c=5.

So, coordinates of FOCI are (±c,0) i.e. (±5,0).

Right choice is (c) 5 units

Explanation: Equation of circle with center at x-axis (a, 0) and RADIUS R units is

(x-a)^2+(y)^2=r^2

=>(2-a)^2+(0)^2=r^2

And (0-a)^2+(4)^2=r^2

=>(a-2)^2=a^2+4^2 => (-2)(2a-2) =16 => a-1=-4 => a=-3

So, r^2 = (2+3)^2=5^2

r=5 units.

Right choice is (c) 8 UNITS

Easy explanation: COMPARING the equation with \((\frac{x}{a})^2-(\frac{y}{b})^2\)=1, we get a=3 and b=4.

Conjugate AXIS length = 2b = 2*4 =8 units.

The CORRECT option is (C) y^2-8x-4y+12=0

Easiest EXPLANATION: Since vertex is at (1, 2) and FOCUS (2, 0) so parabola equation will be

(y-2)^2=4*2(x-1)

y^2-8x-4y+12=0.

Right CHOICE is (c) y=-25

To explain I would SAY: Comparing EQUATION with x^2=4ay.

4a=100 => a=25. Directrix is a line parallel to LATUS RECTUM in such a way that vertex is at middle of both.

Equation of directrix is y=-a => y=-25.

Right ANSWER is (a) x=25

The best EXPLANATION: Comparing equation with y^2=4ax.

4a=100 => a=25. Line PASSING through focus PERPENDICULAR to AXIS is latus rectum.

Equation of latus rectum is x=a => x=25.

The correct answer is (d) (2, 5)

Explanation: COMPARING the equation with GENERAL form x^2+y^2+2gx+2fy+c=0, we get

2g=-4 => g=-2

2f=-10 => F=-5

c=4

Center is at (-g, -f) i.e. (2, 5).