Mathematics Test - 11

The interesetion of line and circle is (0, 0) and (2, -2). Now, taking option (c),

Hence, option (c) is the correct answer

Main Concept :
Equations and Parameters of Standard ParabolaStandard equation of the Parabola:

Consider as S be the focus,  ZZ' be the directrix of the parabola and P(x, y) be any point on parabola.

Let AS = AK = a (a > 0) then coordinate of focus S is (a, 0) and the equation of directrix KZ is x = - a or x + a = 0.

If point P (x , y) lies on the parabola then its distance from the focus S and directrix line L must be equal.

Axis of a parabola: A straight line passes through the focus and perpendicular to the directrix is called the axis of parabola.

For the parabola  y² = 4ax, x - axis is the axis. Here all powers of y are even in y² = 4ax. Hence parabola y² = 4ax is symmetrical about x - axis.

We can also generalise this fact that a parabola is always symmetric about its axis.

(2)  Vertex : The point of intersection of a parabola and its axis is called the vertex of the parabola.
The vertex is the middle point of the focus and the point of intersection of axis and the directrix. For the parabola y² = 4ax, A(0, 0) i.e., the origin is the vertex.

(3)  Double - ordinate: The chord which is perpendicular to the axis of parabola or parallel to directrix is called double ordinate of the parabola.

Let  QQ' be the double - ordinate. If abscissa of Q is h then ordinate of Q, y² = 4ah or y = 2√ah (for Ist Quadrant) and ordinate of Q' is y = -2√ah (for IVth Quadrant). Hence coordinates of Q and Q' are (h, 2√ah) and (h, -2√ah) respectively.

(4)  Latus - rectum:  If the double - ordinate passes through the focus of the parabola, then it is called latus - rectum of the parabola.

Coordinates of the extremeties of the latus rectum are L(a, 2a) and L' (a, - 2a) respectively. Since LS = L'S = 2a.

∴    Length of latus rectum LL' = 2(LS) = 2(L'S) = 4a.

(5)  Focal Chord: A chord of a parabola which is passing through the focus is called a focal chord of the parabola. Here PP' and LL' are the focal chords.

(6)  Focal Distance (Focal length): The focal distance of any point P on the parabola is its distance from the focus S i.e., SP.

Here, Focal distance SP = PM = x + a

Note:  If length of any double ordinate of parabola y² = 4ax is 2l, then coordinates of end points of this double ordinate are

Equation of Circle in Complex NumbersEquation of a circle : (i) The equation of a circle whose centre is at point having affix z₀ and radius r is |z - z₀| = r.

Properties of DeterminantP - 1: The value of determinant remains unchanged, if the rows and the columns are interchanged.

Since the determinant remains unchanged when rows and columns are interchanged, it is obvious that any theorem which is true for 'rows' must also be true for 'columns.'

P - 2: If any two rows (or columns) of a determinant be interchanged, the determinant is unaltered in numerical value but is changed in sign only.

P - 3: If a determinant has two rows (or columns) identical, then its value is zero.

P - 4: If all the elements of any two (or column) be multiplied by the same number, then the value of determinant is multiplied by that number.

P - 5: If each element of any row (or column) can be expressed as a sum of two terms, then the determinant can be expressed as the sum of the determinants.

P - 6: The value of a determinant is not altered by adding to the elements of any row (or column) the same multiples of the corresponding elements of any other row (or column)

Then b.

P - 7: If all elements below leading diagonal or above leading diagonal or except leading diagonal elements are zero then the value of the determinant equal to Product of all leading diagonal elements.

P - 8: If a determinant D becomes zero on putting x = a, then we say that x-a is factor of determinant.

P - 9: It should be noted that while applying operations on determinants then at least one row (or column) must remain unchanged or, Maximum number of operations = order of determinant - 1

P - 10: It should be noted that if any row (or column) which is changed by multiplied a non - zero number, then the determinant will be divided by that number.

Main Concept :

Important terms in Ellipse

Since, we are given the equal intercept of the coordinate axes i.e. x=y=z=p

Therefore, it make a cube

Main Concept :
Section Formulae in 3D1 Internal Division
Let P(x₁, y₁, z₁) and Q (x₂, y₂, z₂) be two points. Let R be a point on the line segment joining P and Q internally in the ratio m : n. Then, the coordinates of R are

2 External Division
If P and Q are such that R divides the join of P and Q externally in the ratio m : n. Then, the coordinates of R are

Main Concept :
Multiplication of matricesTwo matrices A & B can be multiplied only if the number of columns in  A is same as the number of rows in B.
Properties:
(i) In general matrix multiplication is not commutative i.e. AB ≠ BA .

(ii) A(BC) = (AB)C [ Associative law]

(iii) A.(B+C) = AB + AC [ Distributive law]

(iv) If AB = AC ⇒ B = C

(v) If AB = 0, then it is not necessary A = 0 or B =0

(vi) AI = A = IA

(vii) Matrix multiplication is commutative for + ve integral i.e. A^m+1 = A^m A = AA^m

Other Concepts :

Two matrix A and B are said to be equal matrix if they are of same order and their corresponding elements are equal.