Locus MCQ Quiz in தமிழ் - Objective Question with Answer for Locus - இலவச PDF ஐப் பதிவிறக்கவும்

Let be the point on the locus.

Then by the given conditions,

Also, since lies on the given locus, therefore

Comparing Eqs. (i) and (ii), we get

Let ,

,

,

Calculation

sin²y = cos²x

siny = ± cosx

If sin y = cos x = sin( - x)

⇒ y = nπ + (-1)ⁿ( - x)

If sin y = -cos x = sin(x - )

⇒ y = nπ + (-1)ⁿ(x - )

Hence option 3 is correct

Given:

The ratio of distances of point P(x, y) from point (1, 1) and line x − y + 2 = 0 is 1 : √2

Concept:

Use distance formulas to relate point-to-point and point-to-line distances.

Formula Used:

Distance between two points: √[(x₂ − x₁)² + (y₂ − y₁)²]

Distance from point to line ax + by + c = 0: |ax + by + c| / √(a² + b²)

Calculation:

Let P(x, y) be the variable point

Distance from P to (1, 1): √[(x − 1)² + (y − 1)²]

Distance from P to line x − y + 2 = 0: |x − y + 2| / √2

Given ratio = 1 : √2

⇒ √[(x − 1)² + (y − 1)²] = (1 / √2) × |x − y + 2|

⇒ √[(x − 1)² + (y − 1)²] = |x − y + 2| / √2

⇒ Square both sides:

⇒ (x − 1)² + (y − 1)² = (x − y + 2)² / 2

⇒ x² − 2x + 1 + y² − 2y + 1 = (x² − 2xy + y² + 4x − 4y + 4) / 2

⇒ x² + y² − 2x − 2y + 2 = (x² − 2xy + y² + 4x − 4y + 4) / 2

⇒ Multiply both sides by 2:

⇒ 2x² + 2y² − 4x − 4y + 4 = x² − 2xy + y² + 4x − 4y + 4

⇒ Move RHS to LHS:

⇒ (2x² − x²) + (2y² − y²) + (−4x − 4x) + (−4y + 4y) + 2xy + (4 − 4)

⇒ 3x² + 3y² − 8x + 2xy = 0

∴ The equation of the locus is:

3x² + 2xy + 3y² − 12x − 4y + 4 = 0

Calculation

Equation of LM:

y - 2 = (x - t)

⇒ M(0, )

Midpoint of LM: N(h, k)

⇒ 2h = t, 2k = 4 -

⇒ x² = 2 - y

Hence option 2 is correct

Explanation:

∴ The locus of point P will be a straight line which is the right bisector of AB.

Concept used: 

Angle subtended by the diameter of a circle at a point on the circumference is Right angle (90°).

.

Explanation:

Join AB and draw a circle with AB as diameter. The locus of point P is the circumference of this circle, since Angle subtended by the diameter of a circle at a point on the circumference is Right angle (90°). Therefore, whatever be the position of point P, the measure of ∠ APB will always be 90°.

∴ The locus of point P is the circumference of the circle with AB as diameter.

Concept:

The distance of a point (h,k) from a line Ax + By + C = 0 is 

The distance between two point (x₁,y₁) and (x₂,y₂) is 

Calculation:

Let the moving point be (x,y)

then its distance from (4,0) is

and the distance of the point (x,y) from the line x = 16

that is, the distance of the point from the line x + 0y - 16 = 0 is 

⇒ 

Given that, the distance from the point (4, 0) is half that of its distance from the line x = 16

⇒ 

Squaring both sides,

⇒ 

⇒ 4(x² - 8x + 16 + y²) = x² - 32x + 256

⇒ 3x2 + 4y2 = 192

∴ The correct option is (1).

Concept:

Locus: The locus is the collection of all those points whose position is defined by a particular condition.

Internal sectional formula: Let P (x₁, y₁) and Q (x₂, y₂) be the endpoints of the given line segment PQ and R(x, y) be the point which divides PQ in the ratio m: n.

Then, the coordinates of R (x, y) are .

Explanation:

Let R(x, y) be the mid-point of OM and (α, β ) be the co-ordinate of the point M. Then

x = and y = 

⇒ α = 2x  and β = 2y

Therefore, the coordinates of M are (2x, 2y).

The slope of OM = 

and Slope of MP =

Since OM is perpendicular to MP. Therefore

Slope of OM × Slope of MP = -1

⇒ 

⇒ y(2y - b) =  - x (2x - a)

⇒ 2y² - by = - 2x² + ax

⇒ 2x² + 2y² - ax - by = 0

Therefore, the locus of the midpoint of OM is 2x2 + 2y2 - ax - by = 0

Concept:

Distance between (x₁,y₁) and (x₂,y₂) is 

Calculation:

Let the coordinates of P be (x,y)

Given: PA + PB = 3 where A = (1, 2), B = (2, 1) 

⇒  +  = 3

⇒  = 3 - 

Squaring both sides,

⇒  = 9 +   - 6 

⇒ x² - 2x + 1 + y² - 4y + 4 = 9 + x² - 4x + 4 + y² - 2y + 1 - 6

⇒ 5 - 2x - 4y  = 14 - 4x - 2y - 6

⇒ 6 = 9 - 2x + 2y

Again squaring both sides,

⇒ 36[] = (9 - 2x + 2y)²

⇒ 36[x² - 4x + 4 + y² - 2y + 1] = 81 + 4x² + 4y² - 36x + 36y - 8xy

⇒ 36x2 + 36y2 - 144x - 72y + 180 = 81 + 4x2 + 4y2 - 36x + 36y - 8xy

⇒ 32x2 + 32y2 + 8xy - 108x - 108y + 99 = 0

∴ The correct answer is option (3).